Regarding silence:
Musical instruments can not be used
Gadgets can be used
Bishops *can* cast Alchemist spells while silenced, even though that ability is not listed as a special ability for the class. Note that *only* Alchemist spells can be used this way. (I like to make a point of having my Bishop cast a spell when she is silenced just to prove that she can.)

Regarding anti-magic zones:
As a rule, spellcasting items can't be used in anti-magic zones.
Musical instruments can't be used in anti-magic zone.
Potions that cast Booze can be used. Other potions, however, can't. (Does it make any sense that you can't drink potions in anti-magic zones?)
I am pretty sure that bombs can't be thrown, though I don't think I've ever tested that. I also haven't tested the Flamethrower (which I am guessing can't be used, even though it would make sense for it to be usable.)
Gadgets *can* be used. This is, in fact, the only way to heal (other than resting or regeneration) in an anti-magic zone. (Of course, such zones are so small it is easy to just leave the zone to heal.)

A few targeting quirks:
Radius targeted spells can hit enemies that are out of sight. The spell must be centered somewhere visible, but it will hit enemies in the radius that are not visible. (Sometimes Fireball can hit an enemy that can't be hit with Nuclear Blast, for example.)
Group targeted spells require one enemy in the group to be in range (not necessarily in sight), but will hit the entire group, including enemies that are out of range, if it can hit at least one.

A few spell quirks:
Heal Wounds and Heal All do not appear to be capable of backfiring. Stamina and Haste can backfire, however. (Does backfired Haste target Fire or Water resistance?)
Guardian Angel prevents *all* damage taken while in effect. This includes things like poison damage.
Noxious Fumes can't cause nausea in androids, but can still (if you're lucky) knock them unconscious.
Undead are immune to all instant death, including Quicksand. Conversely, flying creatures have no special immunity to Quicksand and Earthquake.
Bat Vampires, even though they look like other Bats, count as undead. This includes being susceptable to certain spells.
Stamina and Rest All don't count as practice during combat, but (unlike spells like Bless) will count during fake combat (starting combat when there's nothing to fight).
Power Cast does affect Identify Item, which has the effect of reducing the cost to identify an item.
Power Cast, I believe, affects Light's duration, even though the power level doesn't (I should double check this once my Bishop learns Power Cast).

Class ability quirks:
Turn Undead exists as a non-learnable spell. It hits all undead in range, but has only Thrown range, for damage + some status ailment (IIRC paralyze).
Pray for a Miracle can produce various effects. One of them is Magic Missile, which is not normally available to a Priest. It can also mimic Turn Undead or heal.
Bishop's Remove Curse ability will fail if the target doesn't have any cursed items.

A few more:

If the entire party is blinded, the enemies will turn invisible and not appear on the screen. (From what I have read, if someone in your party dies, that character is no longer blind and enemies become visible again. Does that make any sense?)

If battle ends, any enemy that is blind will not see you outside of battle. You can go right up to the enemy without starting combat. If you do start combat, it seems the enemy will attack instead of running away the way blind enemies usually do.

In Umpani headquarters, I noticed when of the Umpani referring to my all-female (except RFS-81) as "girls". I think that was a nice touch. (Of course, from what I have read, in version 1.0.0, an all female party will have trouble later in the game; the fix, of course, has the side-effect of having lesbian sex produce an offspring.)

More on anti-magic zones:
Potions fail, but drinks will work. This includes that Cheiftan's Malt item that casts Cure Lesser Condition and the Doubleman Drought that casts Superman.
Holy Water (the item) does not work.
Bread works, but Pickmeup Potions do not (even though they both cast Restore Health).

Very nice insights.

A similar topic: Underwater areas. My understanding is that spells still work, but none from the Fire realm. Oddly enough, I THINK this applys to instruments as well - so instruments still work unless they "cast" fire spells (I haven't tried all of them, but Arresting Area (Freeze All) worked, Rousing Drums (Haste) not). Does anybody know about gadgets?

I *think* the same rule applies to gadgets underwater, but I don't think I've actually tested it. I am a ways away from going underwater with my current party, unfortunately. (There are at least 4 gadgets that are in the fire realm: Lightning Rod, Lava Lamp, Strobe Light, and the one that casts Boiling Blood.)

More anti-magic tests:
The Tincture of Shadows does work.
The Flamethrower does not (even though it would make more sense if it did).

The Flamethrower is unusual and (I believe) unique: it is not a weapon, but it uses the Modern Weapons skill anyway. Unfortunately, it only casts Firestorm at power 1, and Firestorm is one of those spells that is not useful at low power levels.

The Anti Magix Area probably only blocks spells form casters and specific item types, while gadgets and drinks etc work.

The underwater flag probably supresses all fire related spell effects, regardless of item type.

Power Cast also effects duration of buffs, visible when you hover over the spell symbol on top of the screen,
also helps to breach the targets magical resistance in the related realm, i.e. increased damage from spells.

Hold the right mouse button and push to lift the target cursor and place it on otherwise inaccesable spots like bridges, mountain slopes etc.
Note that the game has invisible physical walls in many places which may prevent that.

When you started to move in combat, you can not select another action with your PCs,
but as long as you dont hit the X to end movement, you can redirect the selected actions, incl spells, to another target.

You can target unseen enemies with spells that have the square symbol.
Select spell, power level and hit [N] to cycle through all possible groups, incl friendlies!
Active X-Ray is highly recommended, as the targets dots will blink in radar.

Fyi dragons can spit acid in anti-magic areas

That you can cast Alchemy spells while silenced isn't a class-specific perk per se, being a trait of the Alchemy spellbook instead, and thus applicable to all Alchemy casters.; it's only listed as a feature for the Alchemist because it distinguishes that class from the other three specialist casters.

While only fire spells and effects are blocked in the underwater zones, it's worth bearing in mind that most enemies in that area will have high Water resistance, and many are also heavy on the Mental resistance.

IIRC, you can swipe Anna's stun gun, but she will call you out over it later (and turn hostile). As far as I know this is the only case where someone will react to your pickpocketing even if you aren't caught at it.

Another one regardings spells:
The power level for Summon Elemental is restricted by the surrounding area, i.e. by how much space there is for the elemental. For example, in the retro dungeons you can only cast Summon Elemental at power level 3, in Mt. Gigas Caves (I think) power level 4.
This, however, does not apply to the Canned Elemental potion, which will always be at level 6 no matter where you use it.