Several questions here. I could find answers for casting in armor, but not while using shields or held objects like weapons. The rules say the spellcaster must be able to move his hands, but this is ambiguous to me.
Can a magic-user cast spells while he has a shield equipped?
1a. If yes, does the magic-user still benefit from the shield’s AC bonus that round?
1b. If yes, can he cast with a shield equipped and a weapon held?
Can a magic-user cast spells while holding just a weapon or other object in one hand, leaving the other unoccupied?
2a. If yes, can he cast with both hands holding such objects/weapons?
Do the answers to any of the above questions change when a character who can normally cast in armor (divine spellcaster, Elven Spellsword, etc.) attempts to do so? If so, how?
An arcane caster can "equip" a shield but unless he knows how to fight with a shield, he will gain no AC benefit from doing so.
An arcane caster must have at least one hand free in order to cast spells. Therefore...
1. Can a magic-user cast spells while he has a shield equipped?
APM: Yes, provided he has a hand free.
1a. If yes, does the magic-user still benefit from the shield's AC bonus that round?
APM: Yes, provided he knows how to use shields in the first place.
1b. If yes, can he cast with a shield equipped *and* a weapon held?
APM: No.
2. Can a magic-user cast spells while holding just a weapon or other object in one hand, leaving the other unoccupied?
APM: Yes.
2a. If yes, can he cast with both hands holding such objects/weapons?
APM: No.
3. Do the answers to any of the above questions change when a character who can normally cast in armor (divine spellcaster, Elven Spellsword, etc.) attempts to do so? If so, how?
APM: Ability to wear armor and ability to use a shield are distinct.
My group tends to flaunt this rule, but should you chose to play with it, I would allow a spellsword to drop their weapon for free should they want to cast (you can pick it up next turn or after the battle, but it does help if you have a back-up weapon to draw). This is one of the annoying thing about caster-fighter hybrids: you only have so many actions and so many hands. Think of how bad it is for clerics or dual-wielder blade-dancers. If wielding a two-handed weapon, you could hold it (though not use it) with one hand while casting.
That's technically correct. However, for a caster wielding two one-handed weapons, I rule the caster can hold both weapons in one hand for one round without having to lose an action sheathing his weapon(s) or having to drop them. For a caster with weapon and shield, I would let him hold his weapon in his shield hand for one round (no AC benefit from shield while doing this).
It's up to you how strict you want to be with this.
I tend to take this sort of lax ruling. It really isn’t that hard to shift objects to free one of your hands for 10 seconds or so. This gets more complex, however, if you are lacking in functional hands, or if the hand you wish to cast with is holding something that is on fire.