Definitely. You can pry open the inside lid (the one with the hole in it) after you're done, then put your new powder in.
No, not refillable, not without lots of creative effort, active tinkering, and mess. But this product lasts *forever*, so little do you need for each application. By the time you run out, they'll probably have introduced a new, equally cute special edition design!
Also, if you are willing to stick with this same Tarte powder, have you considered just transferring the lid when you run out and get a new, regular edition jar? I think that might be feasible. It should screw on to a regular version of the same product. I think the jar dimensions are the same, just decorated differently. It's a possibility worth looking into. I don't have one of each to compare; I'm just thinking about the logistics and likelihood.
Yes! You can remove the inside packaging for it and put something else inside.