Is the new framing along the exterior walls built in front of an existing, finished exterior wall? Will those open exterior wall stud bays be insulated? 

I would think the 10" flex duct could transition into square duct above the closet ceiling, then run inside the stud bays, then transition back to flex beneath the floor. Idk which way the floor joists run but it seems doable either direction. 

This looks like it was just run whichever way was easiest for the installer and in the absence of a plan. It happens, can certainly be tightened up

I would move it into the bedroom.

I don't like weird chases and soffits in the corners of rooms. I like clean rectangular rooms. I would try to do it better in the closet.

It looks like if you leave it in the closet, you might be able to build it in and put shelving next to it. If that's the case, I'd leave it in the closet.

Our contractor thinks he can fit in a shoe box shelf next to it, so we’re going to leave it. Thanks for the feedback!

IMO better where it is in a kids room than taking up bedroom space .