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.

The gargantuan animated object continues to be animated.

In this tweet Crawford claims that:

No longer being a valid target trumps condition carryover.

So whether a gargantuan animated object continues to be animated, hinges on what a valid target is for the animate objects spell. The relevant limitation is this:

You can't animate any object larger than Huge.

However, the animate objects spell also says that:

Each target animates and becomes a creature under your control until the spell ends or until reduced to 0 hit points.

Moreover, as Slagmoth explains in this answer:

Nothing is considered an object and a creature simultaneously.

Therefore, enlarging a huge animated object doesn't make it an invalid target of the animate objects spell, because you enlarged a creature, not an object.

In other words, to make the target of the animate objects spell invalid, you must enlarge the object. However, that's impossible to do for the duration of the animate objects spell because there is no object for you to enlarge.