Yes, absolute zero is the MINIMUM energy state for an object, not a ZERO energy state. The subatomic particles would all be in their lowest energy states, but for some that state would still entail some motion I believe. I'm sure the uncertainty principle matters here somehow.
Well, if I had an object at 0 Kelvin (strictly speaking, some non-zero temperature that is very close to zero) I could still move it around my lab... So the answer has to be yes.
If you were asking about internal movement of the particles making up the object, then Drakkith has already answered.