In Wizardry 8, I have noticed that a few templates seem to hold true, almost universally.

For non-casters (Fighter, Rogue, Bard, Gadgeteer, many hybrids), the rule seems to be: Strength and Dexterity, then Speed and Senses. This is because Intelligence is mostly worthless as a whole, and Powercast won't benefit a noncaster. Piety doesn't increase mana for a noncaster, only stamina, which can be gotten elsewhere (Iron Will can be substituted by Magic Screen, Element Shield, and Soul Shield). Vitality increases hit points, but it benefits the Fighter the most, and Mage the least, and physical attacks won't do you in late game, it's usually magic, and Iron Skin isn't that great, and is very hard to train.

For specialist casters (Priest, Alchemist, Psionic, Mage), the rule seems to be: Intelligence and Speed, then Piety and Senses. This is because Piety will actually help the one-spellbook casters with spell points, who really need it. Strength and Powerstrike seem like less of a concern, considering how seldom they will use these (only against bosses). Vitality is minimized for the reason above, and Dexterity is suited more for physical damage characters (Reflextion shouldn't be an issue, as most of your specialist casters shouldn't be on the front lines).

For Bishops, the rule seems to be: Intelligence and Speed, then Strength and Senses. This will grant them Powercast and Snakespeed first, then Powerstrike and Eagle Eye later. Piety is a non-issue for a Bishop with access to every level 1-5 spell, and most of the good level 6 and 7 spells. Vitality and Dexterity ignored for the reasons above.

I would disagree here.
* Speed isn't something every caster will want, particularly if you're playing on phased. Thing is, sometimes going later in the round can be benificial, like if you're casting Heal All (heal damage received after the start of the round) or Purify Air (to remove that Death Cloud (that hasn't been cast yet as of the start of the round) before it has a chance to work.
* It is certainly possible to use casters in the front line, especially Priests, and such characters may want stats like Strength and Dexterity. In particular, Dexterity's expert skill is more likely to be needed by a specialist caster (who is quite frail) than by a Fighter (who doesn't need the extra AC, even in the front line).
* Some players don't even max out Intelligence for Priests, seeing as the need to penetrate enemy resistances aren't as strong. (Do, however, remember that Power Cast benefits healing and support spells.) What about doing Senses earlier in order to get Eagle Eye?

(Rule of thumb: If a stat isn't one of the first two you train, expect to not get the expert skill before it's too late to train it.)

There are many exceptions. There are Battle-Priests and Battle-Bishops. There are Bards and Gadgeteers who only use their stamina-magic, without fighting.
As a whole, especially melee characters like the Fighter and Rogue, and most hybrids (who should be focused on their primary physical skills before magic), the focus should be on physical damage, hence Strength and Dexterity.
I prefer my casters to focus on Speed and Senses so that they always act first, Speed first for Snakespeed. Doing Senses first can work as well, as I still get the same initiative bonus, while gaining Eagle Eye.
The thing about going last is, you don't know what the enemy is going to do. Casting Heal All or Purify Air at the end of the round, if the enemy doesn't do any damage to your party or cast any cloud spells, is a wasted round.
A Bishop or Priest can be on the front lines, but I consider them subpar, as compared to a Fighter, Rogue, or hybrids, who are much tougher, and who are designed for melee combat.
My Valkyrie and Ranger, for the reason you mentioned, are minimizing Intelligence and Powercast, as they have the two best defensive spellbooks (Divinity and Alchemy).

(Assuming phased, not continuous, mode)

Thing is, however, that:
* If you start the round at full health, a fast Heal All isn't going to do you any good, whereas a slow Heal All will heal some of the damage you take that round.
* If you cast Purify Air before the enemy casts Death Cloud, then you risk having characters die at the end of the round. On the other hand, if you cast Purify Air *after* the enemies have cast Death Cloud, and it's cast at a high enough power level, then the Death Cloud won't have a chance to affect anyone.

(By the way, how does Str/Vit for a dwarven battle Bishop sound? The idea is to get extra HP and damage resistance (which is affected by a dwarf's Vitality) in order to keep the character alive on the front lines, and will eventually lead to Iron Skin, probably at level 16 or so.)

Having a Dwarven Battle Bishop sounds great, in theory. You would have 100 Vitality, and unlock Iron Skin (training that is a whole process that I won't get into, try it at Marten's Bluff, where the dead T'Rang is with the arrow trap, where you get the T'Rang Arm). At max (100) Iron Skin skill, your character would have 30% damage resistance.
You should have a character in your party capable of casting Body of Stone (Alchemy only spell), that would add 70% damage resistance to said character, making them immune to physical damage. Keep in mind, however, that they still need magical protection (or so I believe), so casting Guardian Angel on them will prevent negative status effects, I could be wrong, why don't you try it out yourself?
It could be that having 100% damage resistance means that, since no damage is dealt, no status debuffs will occur. Somebody definitely needs to check out a character with maxed Iron Skin and power level 7 Body of Stone cast upon them.
I'm pretty sure that the only time you have to deal with cloud spells guaranteed is with the Djinni of the Clouds battle just before the Chaos Temple. Other times, enemy spellcasters vary with their spell picks, and you can always reload if you don't like the outcome.

Because you can't save during combat, doing that can mean losing a lot of progress if it happens later in a longer fight.

Also, there's Ironman mode, which disallows such reloading unless you force kill the game.

This is why I don't play on Ironman Mode. There are way too many (sometimes unfair) variables that can happen. This game should be based on skill, not RNG.
It goes without saying that the Alchemist, Psionic, and Mage have poor one-handed weapons (daggers, wands, and short staves), and the last two don't even get the Shield skill.
Therefore, of the casters, only the Bishop and Priest stand as worthy melee characters, getting the Mace and Flail skill, in exchange for sacrificing daggers (worth it).
The Bard and Gadgeteer are middling characters, jack of all trades, master of none, which is why I recommend Sword and Shield for them (their damage potential isn't good enough to warrant the penalties of Dual Wielding).
The Fighter can take maximum advantage of their melee abilities (Close Combat bonus, KO% in both melee and range, stamina regeneration, Berserk attack option), by Dual-Wielding. Choice weapons would be Sword, and Mace and Flail (that way, they could get both *Light Sword* and Diamond Eyes).
The Rogue is universally agreed to dual wield, using Sword and Dagger skills. The only debate here is whether they want to be cursed with Bloodlust and Thieves' Dagger, or uncursed with *Light Sword* and Stiletto.
The Valkyrie is pointed towards Polearms, due to her profession bonus. No argument here.
The Ranger can do many things, however, in this party, they are a flank character, so I'm going Polearm, so they can hit from extended range their own flank, as well as the front.
The Bishops are far too busy with training magic, to worry about physical damage. This will change later, once magic skills are maxed.
The other classes (not to be judgemental, as I haven't tried most of them), seem subpar.