I have been having trouble at this particular spot in Morrowind, and in the corresponding spots in Daggerfall and Oblivion (and Battlespire). Could anyone help here?

I will begin by saying that it is very effective in all three games to use a weapon backed up by a little magic.

I will also say that Long Blades are by far the most common-sense approach if you want to have an easier time. In Daggerfall, as far as I've seen, they drop in much more plentiful quantity than the other weapons and sport nice, balanced damage rolls. In Battlespire, lore-wise, Long Blade and Spear (counts as LB) are both important to the story. In Morrowind, it matters a bit less, but it is somewhat similar to Daggerfall in quantity and effectiveness of Long Blades vs. other weapons.

The rest of this is assuming you want a balanced character who can complete all the content without too much of a struggle.

MORROWIND:
I'll start with Morrowind, since you have included a screenshot of the game. To be completely honest, the game becomes much easier if you just focus on one weapon. Just get that hit chance high enough, get a few nice defensive/utitlity enchantments, and it is truly enough. However, I am adding a prototype below with some empty/multi-choice skill slots to play around with.

Major:
1. Weapon Skill (Long Blade and Spear are the easiest. Spear has distance and raises endurance (helpful). Long blade mentioned above.)
2. Armor Skill
3. Utility or R Skill (All these are listed below)
4. Utility or R Skill
5. Any. This is a very good place for Restoration or Destruction.
Minor:
1. This is a good place for Security or Alteration for lockpicking.
2. This is a good place for Sneak unless taking Illusion.
3. This is a good place for Speechcraft unless taking Illusion.
4. Mysticism (If in Major, sub a Utility)
5. Alteration (If in Major, sub a Utility)

Recommended Skills: Alteration (lockpick/levitation), Mysticism (teleportation)
Utility Skills: Block, Armorer, Mercantile, Speechcraft, Illusion (can replace previous two + sneak)
Complementary Skills: Restoration (healing), Destruction (magical damage)

DAGGERFALL:
Daggerfall is a little different. Weapon choice isn't as be-all-end-all, but it still does help. Using the mouse-upward forward/stabbing attack makes it a bit easier to hit, even with weapons you have nearly no skill in. However, the damage will suffer. This is a good way to get around weapon immunities if you have, say, an Ebony weapon lying around that you are unskilled in and your Elven Mace isn't cutting it.

Arrows have weight, gold has weight. Bows can backstab through doors.
1. Weapon Skill
2. Critical Strike
3. Backstab, if you use Marksman. Dodging, if you don't manually dodge. Otherwise, Utility or Magic.

1. Mysticism (Recall out of labyrinthine dungeons, helps with time limits)
2. Thaumaturgy (Levitation, Spell Reflection)
3. Alteration (Survive in water)

All the rest: Whatever you want. Sounds cool? Cool. Add it. You can move Mysticism downward and something else up, Mysticism just needs to be high enough for Recall.

Advantages and Disadvantages: There are a couple well-known cheats here, which I will avoid in recommending a beginner build. You'll probably want some extra Hit Points, so put some points in there first, if so. The "Dagger" will raise. You'll want that as close to the middle as possible, if you want to level at a decent rate. I'm goint to recommend what I found fun/useful/not so detrimental below, and you can balance as you see fit.

Advantages:
Increased Magery: Excellent.
Athleticism: Good if you don't want to rest for Stamina as often (or are playing a fast-hitting build such as Dagger or Hand-to-Hand).
Expertise In: Good, adds (level/3) damage as well as hit chance.
Immunity: Paralysis is great. You don't need it if you are playing as a High Elf, however.
Spell Absorption: Can be overpowered. Can also get you killed if you haven't spent enough Magicka to absorb the enemy's spell.

Disadvantages:
Critical Weakness: I never had a problem taking Poison, others vary.
Damage: From Holy Places is easy to avoid, even on non-vampires. Temple services are easy to go in, get, run out.
Darkness-powered Magery: Doesn't hurt much if you're inside dungeons all the time. Hunters and quest-related harassment can show up outside, however. Iffy.
Forbidden Armor: Flavor. Up to you. I went through the game "naked" without much issue, since manual dodging is possible.
Forbidden Material: Silver and Orcish aren't very common.
The rest are pretty self-explanatory or similar in nature.

BATTLESPIRE:
It is certainly something. Early on, the spell-spamming ultra-nuke enemies can be a nightmare, even to themselves. An interesting factor of Battlespire, however, is that if you take immunity in a damage-type, the enemies stop casting spells which deal that type of damage. Every encounter becomes a melee encounter, and the combat becomes very similar to Daggerfall (hit, back up, hit). However, melee enemies can be -more- dangerous in some cases later on. Especially where lava is involved (melee hits tend to knock back a lot further than magic, at least on my PC).

Build is similar to Daggerfall. Long Blade includes Spears, Missile Weapon includes Javelins (which are melee). There are no shops, so buying arrows is not an option. Always keep a backup weapon, and always check empty box-like openable containers to see if they are Coffers of Restoration (to repair your favorite items).
1. Weapon
2. Restoration (for Cure Health)
3. Critical Strike
4. Dodging
5. Hand-to-Hand (incase weapon breaks, unless you chose it as primary weapon)

The rest are not so important. Choose what looks enjoyable.

Advantages/Disadvantages
First spend a few points on a weapon of your favored type in the Equipment section, then...
Armor type affects that type -and all lower types-, so keep that in mind. Regeneration of Spell Points is very good and important. Otherwise, you need to rely on finding blue floating gems throughout the levels (and they are often not enough). Increase Magery is excellent, as in Daggerfall. The rest can be adjusted as you see fit. Make sure to put any excess points in Wounds (Hit Points).

What kind of trouble do you have?

The problem is this:

There are typically a few skills I intend to use during my playthrough. Choosing those skills is not a problem, generally. However, once I have chosen those skills, there are still more slots that need to be chosen, but I don't have any good ideas of what to put there.

I think part of the problem is having to make so many choices at this point, and the games provide too many skill slots to put skills in. This is actually the part where I have trouble.

Anyway, since you took a while to post a nice detailed reply, here are my responses to some of the points: Illusion doesn't help with lockpicking. Alteration, however, does. (Or you could just Enchant an item, which is my preferred option; I love Morrowind's Enchanting, and am disappointed that Olbivion doesn't allow custom "On Use" enchantments.) In Daggerfall, my experience is that this advantage only works against spells you cast yourself (area at range targeting), or maybe against enemies lower in level than you (rare because of level scaling). Therefore, I consider this advantage only useful if you plan on using it to absorb your own spells, which feels like a major exploit.

(This behavior is probably a bug. Probably not the first bug of this type in the TES series, and definitely not the last (in Morrowind, the Blind effect increases accuracy, and Armorer works better at low fatigue).) How did you find out this bit of information? (I happen to like to study the games I play, learning the exact game mechanics, so it would be nice to find out more.)

I happen to like Spell Absorption in Battlespire. This way, enemies will waste their magic casting spells on me, and if I take too much damage, I can just cast Cure Health.

I have noticed that, when using Hand-to-Hand, I don't get knockbacked much, and I also don't have to worry about being disarmed (how would that work, anyway?). Also, once getting familiar with the start of the game, I find that I can just start with 5 in Hand-to-hand, use it as my primary weapon, and have the skill get pretty good in the first level. (Before getting familiar with the first level (in particular, where the first healing crystal is), you could start with 60 Hand-to-hand, which will make the start of the game really easy, and then restart once you know where the healing crystal is.)

Incidentally, the whole thing about knockback only being an issue when wielding a weapon reminds me of a quirk of Oblivion; if you don't have your weapon drawn, you can't be staggered. This is *very* useful when fighting enemies with strong melee attacks; just sheath your weapon and use Descruction magic (or Absorb Health) instead. I find that this happens if I get Spell Absorption from another source (such as an item or spell), but not as a class advantage. Perhaps your level is higher than that of the Ancient Liches?

Incidentally, with Spell Absorption as a class advantage, I *always* absorb my own spells, provided I don't have another Spell Absorption effect active (which seems to override the class advantage). As I said, I really suspect some inverted logic (this doesn't work in Battlespire, for example).

I did make a monk character a while back, so this will be fun to test out. It's an odd difference. She's level 16, and the room (below) was in Greensly Hall. At full magicka, she just falls flat. I put my old save in and went back to the Hall a moment ago, and an Ancient Vampire cast a spell, which she also absorbed. She is wearing Ebony Mail. The rest is self-enchanted, but does not boast Spell Absorption. Edit: I did it again completely naked and pre-cast "water walking" a ton of times to make sure she didn't regain from un-readying a spell. She absorbed his spells.

I might decide to do that if I can't decide. I do, however, reserve the right to reject skills I don't like (Daggerfall's Stealth skill, for example).

Anyway, I want to give Morrowind another go, and my character is currently looking something like this:
Race: Orc
Gender: Female (this is not going to change)
Birthsign: Lady (I have played with Atronach before, but I feel like doing something different this time)
Class Attributes: Endurance (more health), Luck (harder to increase by leveling up)
Specialization: Magic (because magic skills are harder to increase normally)

Major Skills:
Axe (has strongest bound weapon; Bound Battle-Axe can be my primary weapon)
Conjuration (some spells unavilable for custom enchantment, also helps with Bound Axe early, and lets me start with Summon Ancestral Ghost (great for filling soul gems, also great for Atronach characters))
Athletics (faster movement, particularly early on)
Acrobatics (ditto, plus helps against unexpected falls)
Enchant (will be using this skill heavily, and it tends to rise rather slowly, but I want it at a high level for charge cost reduction)

Minor Skills:
Blunt Weapon (has the Ebony Staff and the Mace of Molag Bal)
Mysticism (easier Soul Trap early on, plus needed for House Telvanni ranks)

The remaining Minor Skill slots are what remain. I don't want to put armor skills there, as I'd rather not have my set-up favor certain types of armor over others, and I prefer to tank damage and heal instead of avoiding damage in the first place. Restoration is a possibility (800 point Fortify for 1 second and Cure Blight Disease both require 150+ enchant points, but are reasonable as a spell (though this character may need a helper spell to cast them)).

I like the idea to assign skills randomly, and then find out how the resulting character must be played to be "successful" (whatever this means in your context). Except when you are so familiar with the game mechanics that you are able to play successfully no matter what the skill set is. Have you ever tried to avoid your favourite skills deliberately, to focus on the skills you like least, you find most unsuitable for the respective game?
In the end it always comes down to what you're trying to achieve.

I'm thinking of taking all the remaining melee weapon skills and not taking Restoration. (I thought about omitting Spear, but there's apparently a spear with constant Spell Absorption, and it looks like the master Restoration trainer isn't that hard to get access to (unlike, say, the master Enchant trainer or the master Medium Armor trainer). The main implication of doing so is that I won't be able to get as big a 1 second bonus to a skill early on, but I think I can work with that limitation.

You keep amazing me, lady :-)!
A character like the one you describe must be pretty tough to play, at least initially (and it would be interesting to play it in Daggerfall). There are many skills you can't use at the same time, so they're either unused or they rise slowly. And you have to keep an eye on your health, haven't you? Of course, after a while this character will be quite powerful. Are you planning to play the base game - or to go straight to Solstheim for a nice additional challenge?

[Morrowind]
I have a good strategy for the early game. Start by going to Balmora, return the package for some money, then buy a Bound Axe spell. (With Axe and Conjuration as major skills, this is enough to make me quite effective in combat, even at a low level.) Then rest and kill 2 Dark Brotherhood assassins, equip the armor from one of them. Then go to Caldera, sell the second set of armor to Creeper; this gives me enough money for the early game. Then, I need to get access to Mark, Recall, and Divine Intervention; one trip to Mournhold and I have the Fortify Skill effect. Then go to Tel Branora, and get Common Soul Gems to fill with Ancestor Ghosts; at this point I can start making full use of the Enchant spell (well, I don't yet have access to Constant Effects, but this should be enough for a while).

The skills I can't use at the same time are for later, once I start finding interesting pre-enchanted magic weapons that I want to use. At that point, getting money for training should be a non-issue, or I can fortify the weapon skill.

Makes sense.
If I may ask: When do you finish an individual game? When you have completed a specific objective (like defeating Dagoth-Ur, or rising all skills to 100, say), or when you see no further potential for developing a character and the game gets boring for you?
Most of the time I simply stop playing a (Morrowind) character because I lose interest, normally because the character has become too strong, and I often try to delay this moment by imposing artificial obstacles (like not using certain weapons, armour, spells, etc), and by not using shops or trainers - and most certainly by switching off the assassins, who provide too strong armour and too much wealth in the early game. Different people, different playstiles :-).

Usually, what happens is that I start thinking about another game I want to play, and about said game's mechanics. At this point, I find that I start playing my current game less and less, and want to start up a new playthrough of another game.