1. Can I use the Hardening spell to improve the hardness of an animated object? I know something cannot be magical to be made into an animated object, but could it be applied afterward?
I don't think you could apply it afterwards, as it is now a creature, not an object. However, while you can't animate something that is magical, that is not the same as not being able to animate something that is effected by magic. In addition, the Hardening spell specifically calls out that it doesn't effect other transformations, so I don't see any reason you couldn't cast the spell before.
2. Similarly, could I apply an enhancement bonus to the golem, and if so, what would that give (Enhancement AC, Hardness, HP?)
Not really, although nothing prevents a Golem from using items that grant enhancement bonuses.
3. Can I install armor onto a humanoid animated object, or is that supposed to be represented by the materiel it's made of?
Nothing would prevent an animated statue from wearing armor. It would take non-proficiency penalties though.
4. More overall than anything, but are animated objects considered creatures or objects for spells, I found that they are considered creatures for armor class, but are they otherwise considered objects?
They become creatures of the construct type.
5. Would an animated object's hardness divide cold damage by 4 and all the other bonuses that objects get against certain damage types?
I'm not sure where you are getting divide by 4. The general rules for objects is to divide energy damage by 2 and then apply hardness. I recall some debate about how this works for animated objects, but can't recall the details and I don't know why the general rules wouldn't apply.