[RPG] Can objects animated by the Animate Objects spell take psychic damage

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.

No, the damage is not magical.

From the description of the Animate Objects spell:

Choose up to ten nonmagical objects within range that are not being worn or carried

Let's say the spellcaster animates an ordinary shovel, which can then fly and slam any target. Because this object is nonmagical, there is no reason that the damage should be magical. It is basically as if the caster would throw the shovel to a target.

Also, for example, none of the animated objects from the MM p.19 does magical damage. The Animated Armor and the Rug of Smothering do normal bludgeoning damage and the Flying Sword does normal slashing damage.