Fuzzy-Wuzzy's answer is probably the best technical way to run those situations. I don't like it in all cases myself, I think it leads to potential abuse, but looking at the rules and the writing, that's probably how it should work. I don't rule it that way, but Fuzzy-Wuzzy is probably correct.
I'll tell you why I don't think it should be, not that my way is necessarily 'correct', but since your question has already been answered 'rules-wise' I wanted to share some thoughts.
I tend to view animated objects (not constructs like golems and creature constructs) as such a broad range of things that you can't necessarily have a one-rule-fits-all approach. It's like trying to broadly define an illusion when an illusion can be anything from a smell, to a sound, to a creature, to an object (even then they had to break those up into figments, glamers, what-have-you.)
It comes down to how an animated object is defined. Is it just an object that is now moving (ie. not inanimate) and has just received the Construct type because that is the most appropriate type and 'creatures' need one and also because that creature type spells out the things an animated object is also immune to without having to do up an individual listing for each potential object in existence? Does an animated object become a construct that just gets hardness or does is have hardness because it's still an object (just with a creature type)?
If you decide that an animated object is an Object, then it takes half-damage from ranged attacks and energy attacks before applying hardness. It also wouldn't be subject to magic missiles despite being a 'construct' because it's also an object and that spell can't affect objects. Also, technically an object gets the broken condition at half hit points, and I don't think anyone's ever run such a thing, though it would technically only apply to animated weapons, armor, or shields, anything else wouldn't really have any mechanical penalties (Improved Critical ability might drop to 20 only, but that's a specific case of a specific object with a specific Construction build).
Does locate objects find an animated object if it's the type of thing you're looking for? If you are concentrating on finding your lost sword, does the spell fail because some jerk stole it and cast animate objects on it so now it isn't an object? I tend to say it is.
"Can I sense any books in this library?"
"No, but you are surrounded by books."
I read the James Jacobs post. I mean that's a great simplification of things. It looks good on paper, not good in implementation. I just don't think it adds up. If your party comes up against a door, and they can't smash it down for instance because they have don't have an 'appropriate' bludgeoning or slashing weapon. Suddenly the caster casts animate object on the door and that magic is expected to 'make the door weaker'?
"Now I can hack through this door with my spear or my shuriken!"
"Now magic missiles damage the door even though that is clearly spelled out as not being the intention of the spell or what it should be capable of!
If you're being tied up by an animated rope (not animate rope cause that rope's TOTALLY different), I'm going to rule that your bludgeoning weapon is ineffective at cutting through it. Why is an animated rope, suddenly easier to destroy with a sling bullet than a rope effected by animate rope? Is a rope of climbing a Construct? No, it's an object. Is it an animate object? Yes, can you cut in two by beating it with your warhammer? No.
Are you really going to believe that casting animate object on your candelabra and candles makes them burn and melt twice as fast because they are taking twice as much fire/heat damage as a normal candle? Would an animated block of ice melt twice as fast as an equal size block of ice in all the same conditions, even if the animated one wasn't moving? Of course it wouldn't. (Oh wait I forgot, ice doesn't melt in any case because animate or not it doesn't take nonlethal damage from warm environments.)
"You know that bed-warming pan you've had for hundreds of years in your family? Well, as you're getting older, I thought it would be nice to animate it, so it could move itself back and forth from your bed to the fireplace to warm the coals inside it when they got cold, so you wouldn't have to. It melted, because apparently coals are hot, at least, just hot enough to melt it... for some reason... even though they aren't actually... hot enough to melt it, I mean."
"How is this possible?!"
"Oh, a comet fly overhead in the skies and that's just how bed-pans work now.
Again, just calling it a 'creature' is probably the simplest 'technical' way to handle it. Saying that the animate objects spell is intended to 'make objects weaker' is not the method I would chose to foist off as an answer necessarily. I would just say animated objects are just animate objects, more so than a waterwheel or a barrel rolling down a hill, but still 'objects'. Why are they Constructs? To show that aren't considered 'unattended' objects or that they autofail saves.
Unfortunately, it requires both a mature group of players willing to accept that not everything fits neatly into round or square holes and a DM able to explain it, and that's not something that a Design Team can necessarily control, so it's probably better to go with a 'give a simplistic answer' rather than a more complex one, because unfortunately, with such a broad range of possible objects, there will always be a judgement call involved.
