Spell description for reference

It seems clear that this spell gives objects a mode of movement which they didn't have before. What kind of movement does it give? Can objects walk, slide, roll, fly, etc.? What if the object doesn't have moving parts?

By default, It will have a land movement. How this movement is executed will depend on his alotment of mobile parts. If the original object has no moving parts, it will achieve mobility by other means. (A stone golem, as a example, is created from a statue sculpted from a single block of stone, and yet it gains joints after it is animated). Also, you can add other movement modes when deciding its characteristics.

Do animated objects act independently of the caster? I assume so, since otherwise, the spell would say something about requiring concentration. Are they treated like summoned animals - attacking enemies and/or obeying orders if possible? Can they refuse the will of the caster - do they avoid placing themselves in suicidal (?) conditions?

How do animated objects determine the enemy so that they can attack? Do they have blindsight or some similar ability? How far does it extend? Do they simply divine where they need to go? If so, are there any limits to their ability to divine locations?

The caster select the target these objects will attack at the moment of casting the spell (he can change that target later, as a move action). The animated objects are considered summoned monsters of the construct subtype, so they are subject to the general rules of summoned monsters. Note that, as per the spell description, attacking their target is all the monster that created will do (standard summon spells, for example, says that monster can perform other actions apart from attacking). Likewise, choosing is The caster can choose a different target as a move action, but has otherwise no other form of control over them. This is how they determine who they can attack, it is all detailed in the spell description.

As for how they see, they are treated as summoned monsters of the construct type, and As constructs, they have low ligth vision and darkvision (60ft) by default.

Can animated objects be debuffed in any meaningful way? Can they acquire status conditions aside from "broken" and the like (i.e., conditions that apply specifically to objects)?

Animated objects cannot gain the broken status because they are considered constructs, and thus creatures, rather than items. They can be debuffed, disabled or impaired the same way any you can do with any other creature, barring construct-specific inmunities and resistances. Note also that animated objects created with the spell are vulnerable to dispelling and antimagic.

How are attack and damage determined for animated objects? Is it related to the caster level, ability modifier, etc.? Can animated objects do anything but move, attack and full attack? When full attacking, do objects get iterative attacks?

The link given by Wyrmwood shows the stat block of a medium sized animated object, and its many options. Depending on those options, and after size changes, if applicable (see here), you will have animated object with different characteristics, that may include multiple attacks. Increasing caster level do not advance the creatures, it allows you to animate more of them, or animate bigger ones.

is there anything stopping the Good-aligned 'Necromancer'?

There shouldn’t really be one anyway.

Is there anything stopping you using Animate Objects, with or without Permanency, to affect a bunch of corpses? If so, what? And would that qualify as an Evil, Chaotic, or Neutral act?

I believe corpses are objects, on the basis of the definition of an object (something without a Charisma or Wisdom score) and what it means to have or not have those abilities:

Any creature that can perceive its environment in any fashion has at least 1 point of Wisdom.

Any creature capable of telling the difference between itself and things that are not itself has at least 1 point of Charisma.

Since a corpse is neither self-aware, nor aware of its surroundings, it should not have Charisma or Wisdom scores, and therefore should be an object.

Moreover, a creature (that is, a not-object, since in game terms everything is either a creature or an object), requires being “alive or otherwise active,” which a corpse isn’t generally.

That said, the game never comes out and says that corpses are objects, and several spells that specifically target corpses lack the (Object) notation that they should have (including some that allow saving throws). Finally, a lot of object-targeting spells were most likely not written with corpses in mind. The mending spell, in particular, seems problematic (since it means a cantrip can do what a higher-level spell, gentle repose, does, and that’s not supposed to happen).

Anyway, point is, if a corpse is an object (and I think it is), animate objects seems (oddly enough) legit. The animate objects spell isn’t inherently Evil like animate dead, nor will your animated corpse involve a soul forcibly bound to it or whatever it is that makes animate dead evil in the first place. Depending on your setting, desecrating a corpse may be considered Evil or un-Lawful.

Have you ever heard of this being done?

No, I have not.