I'm curious what self-imposed rules you adhere to when playing these games.

Here are a couple possible examples that you might consider:

For Arena:
Only do the Oghma Infinum quest once. (This quest can be repeated, allowing you to raise your stats all to the maximum)

For Morrowind:
No creating spells that Soul Trap and have another effect. (From what I have read, doing this can make spells permanent, which is what I would consider a major exploit.)

I've started to use self-imposed rules after I had completed the main quest of Morrowind for the first time. For one, I never use trainers and I never buy weapons or armour, except in the shop in Seyda Neen (to make the start somewhat less cumbersome). Finally, I never increase attributes by "training" them (like jumping around or casting spells before sleeping, for no other reason than to get the attributes up), and I don't use exploits. A few examples:

- No weapons, no armour, no spells except healing.
It took me probably an hour or more to kill the first mudcrab and my fingers were numb, but you get the skill for weapon-less fighting up quite quickly, and then fighting becomes gradually easier (you hit more often and harder). At 100, most fights are relatively easy. Golden Saints remain a kind of challenge, but Dagoth Ur didn't take longer than one or two minutes.
I used healing because I didn't want to bother with buying or brewing potions or with sleeping all the time, and my call-sign was Mage, obviously (spell-point factor 1.5).

- An Atronach Mage with armour, but no weapons, fighting with spells only (both offensive and defensive).
Very tough! After completing the main game, I moved on to Tribunal, but gave up somewhere along the way (didn't have enough mana for a specific fight with a very powerful mage, and didn't want to or couldn't go back to a place where I could stock up potions, if i remember correctly). I should still have my last save somewhere (if I manage to find it), and may try to continue this game eventually.

- No fighting.
This was a very odd but interesting experience. I think I got stuck somewhere (main-quest-wise) because fighting was necessary to move on. But sneaking into all sorts of dungeons, temples etc and looting them is really funny.

- No magic.
I think this was my first attempt at playing according to my own rules, and I found it much too easy.

Let me add that my daughter, at the age of about 10, liked Morrowind very much and always played according to her own rules and objectives, mainly because she didn't like violence and fighting. I remember one game where she played an Argonian girl who spent most of the time in the water, swimming, collecting pearls, exploring the rich under-water scenery and swimming away from fishes, ghosts, etc (I think most Morrowind players don't have a clue that they miss this hidden under-water world).
She also played a kind of a witch, collecting herbs and other ingredients, and experimenting with alchemy.

@dtgreene: It doesn't even have to be soul trap, any spell together with fortify your own works, if you miss your target (hit the ground or a wall.

Atronarch Altmer going (almost) pure spell caster is my preferred playstyle.

You'll definitely want to get Mysticism as high as possible, Mark/Recall, Divine/Almsivi Intervention, Reflect, and Absorb Health are all great effects, Absorb Health being overall more useful than the Destruction effects for damage.

The main issue with Atronarch spell caster is that you need to either abuse a specific bug (at least one code patch fixes it) or rely on Alchemy.

Yeah, bug abusing or cheating will be helpful for such characters. I always make a plugin that enables "Restore Magicka" for spell-making and enchanting.

Good advice - I've neglected Absorb Health so far. Alchemy might help, too, but I always had money enough to buy as many health potions as I wanted. And I gave up in Tribunal because I simply didn't have enough potions with me and didn't want to return to (or reload) to a place where I could get new ones. like I said, I must have an old savegame somewhere, and I may give it another try sooner or later.
My main point is to play by my own rules. These exclude the use of cheats and exploits. And I also don't use Soul Trap and Fortify spells. It's not a matter of principle, simply of playing a fun game. If I wanted an easier game, I would use different rules, of course.

Do you ever use those spells separately for their intended purposes (i.e. actually soul trapping an enemy to put it in a soul-gem, for example)?

Well, I haven't played Morrowind seriously for something like 5 years (recently only the TC Swords of The Perithia), and don't remember everything. I know that I have used Soul Trap, but not a lot, and not for a very long time (and I don't remember why not - possibly because I had problems or didn't see enough value in using it).

If you use items with charges, you really can't avoid Soul Trap for too long. And if you want to make your own enchanted items, there really is no way around it due to a lack of easily available Grand Soul Gems with 400 point souls in them (a requirement to make non-charge enchanted items).

Of course with Alchemy you want to stay away from making/using Fortify Intelligence potions during your Alchemy sessions. At least if you want a challenge. Even so the Healing and Magicka potions you can make with 100 Intelligence and 100 Alchemy are still noticably superior to those you can buy/find.

When I'm in the right mood I enjoy playing playing without benefit of bartering. If you need something, steal it, if you can't steal it you can't have it. Thieves and wizards with unlocking & stealth abilities are obviously favored with this restriction, and even the most barbarous warrior will be tempted to dabble. Extending the barter prohibition to cover spell purchase probably makes spellcasters unviable, but maybe that's your bag. Extending it to cover buying repair services (buying hammers is barter, obviously) might keep you poor the whole game through, always scrounging for new equipment and the occasional repair hammer. Extending it to cover paid training obviously condemns you to endless repetitive training exercises or crappy levels, I wouldn't do it. ;)

No - or rather, it's up to you! I never use trainers in Morrowind, and I never do extensive repetitive training either (like continuously jumping instead of walking, or casting all sorts of spells before resting). I choose a role and develop skills accordingly, by using them,

I just get murdered too hard. Guess I suck at Morrowind.

One rule I've thought up for when playing the game: No permanently raising stats or skills above 100.

Another is no using the soul trap glitch, or at least limiting the use of it.

(Alternatively, maybe I could cheat Hircine's Ring and see if I could use it to remove the effects of "permanent" soul trap glitch effects.)

A shame! Perhaps we can help you. Would you mind to explain your issue in a more detailed way? Like what weapons or spells you are using, which class, which opponents you find particularly difficult, how far you have progressed in the game, etc.
You might consider starting with a Breton mage (of whatever flavour, spellsword, battlemage, ...), sign apprentice or mage. Then walk out of Seyda Neen to the west until you get to the guy who falls from the sky, take his sword (quite powerful in the very beginning, do a few quests (for example, find the tax collector, find the place where the guy who has stolen money hides his "treasure". You can do this without fighting, and it gives some money to purchase the fireball spell, hearth heal and some armour. With this equipment, you should be able to survive the first fights with the mudcrabs and other beach vermin, with either the sword or the fireball spell (advantage: you can use it from a distance, disadvantage: you don't have unlimited mana). With this practise you should be able to survive the bandit cave near the silt strider place and the first one or two tombs along the shore to the west. Now you can improve your weapons and armour, if you like, and do what you like, go to Balmora (walk or use the silt strider - I myself prefer walking), or continue along the western shore, ...
There are thousands of other ways to start the game, but I find the one I described very suitable for less experienced people. If you have difficulties with fighting, go looking for mudcrabs and kwama messengers (there is a nice number of them near Seyda Neen - and run from cliff racers, which you also may encounter).
Good luck and have fun!

PS: Even if you don't like fighting, or think you're not good enough at it, you can still enjoy Morrowind, I guess. For example, play a thief, or something like a tourist, who simply travels around in Morrowind, visits all kinds of places - and runs away from all nasties.

Yeah the sparksword from the Icarian Flight guy is a great help. I'm really not the greatest player, but I was mostly joking about getting murdered too hard. In Morrowind the answer to getting murdered too hard is "come back later." However, I seriously don't know how someone can get a skill like mysticism or alteration to appreciable levels without buying training or spamming cheap spells before rest. When I play a mage without training my primary weapon skill always leaves my spell schools in the dust by level 5 if I'm not training. Consequently, magely builds with self-imposed training restriction works out to be a daggermonk for me, which is not my favorite way to play. :'P

My preferred "Welp the last 3 dungeons whipped me hard" response is to go full flower-picker for a few hours and come back exploiting alchemy. But once you've gone for exploiting alchemy self-imposed challenge rules are a little silly, I find.

I will say of the no-bartering challenge that I'm sick of NPCs keeping all their repair hammers in their pockets or extradimensional vaults. Levelling strength heavily is essential just to carry extra shields.