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.
Despite Miniman's size description, the Heavy and Light properties should make a difference. By RAW,
Heavy. Small creatures have disadvantage on attack
rolls with heavy weapons. A heavy weapon’s size and
bulk make it too large for a Small creature to use
effectively.
Light. A light weapon is small and easy to handle,
making it ideal for use when fighting with two weapons.
A classical greatsword, for example, has 1.75m size, which is probably twice as long as a barrel. The DMG's size descriptions lists:
I would separate between Heavy, Light, and other weapons, and give them separate sizes.
Personally, I'd go with medium-size for Heavy Weapons (big ass chunks of metal), and Small-size for the others. I would only go for tiny-size for some Light Weapons (e.g., small daggers).
Animate Objects, by RAW, describes how your items do damage.
It makes a slam attack with an attack bonus and
bludgeoning damage determined by its size. The DM
might rule that a specific object inflicts slashing or
piercing damage based on its form.
I would also say that part of a weapon's damage is based on how it is swung, how it is used. The wizard does not control them like a martial artist, so the weapons do the damage shown in the Animate Objects table. I'd just change the damage type to correspond to the weapon's damage.
Best Answer
There isn't any explicit rule as written on this subject.
Under the animate objects spell description, it explicitly states that they act on their turn. But there is no reference to when that turn happens. The vast majority of people I have seen talk about the answer to this question have supported that the objects roll a collective initiative and act on that initiative. Some people have argued that they go right after the mage, which was my initial impression, but it seems this is a less popular opinion. I have seen the argument made that they should go right after the mage who casts the spell to speed up combat, but from my experience it makes little difference to the speed of combat when compared to them rolling their own initiative.
With that said, the basis of this majority opinion (that the animated objects roll initiative when the spell is cast) is that the objects get a turn, but it doesn't say when their turn is, so what is done according to rules as written for determining the order of turns in a round? Roll initiative. However, with a close reading of the rules for initiative, one would find that there isn't a rule to determine initiative after the start of combat, here is what is written, emphasis mine:
With that being the case, there isn't actually a written rule for determining the objects' initiative, only that the objects' initiative is what determines their order in combat, this weakens the stance of the majority opinion which usually uses RaW as their axiom for determining that you roll initiative for the objects when the spell is cast.
Arguments can be made to what is RaI, but both sides can be just as well supported by doing so.
I would conclude that it is up to the DM to determine when the objects get a turn. I would suggest you talk with your DM about how the spell is ruled.
Possible ways you can determine when the animated objects get their turn, and my experience with them:
They can be given their turn right after the caster
They can roll initiative when the spell is cast
You can rule that the objects get their turn immediately before the mage. If you do treat the spell this way, write it into your homebrew and house rules you give the players at session 0 so nobody is disappointed when they get to using the spell.