If it were my home, I would do as you're suggesting as far from
opposite the supply vent as you can get it. Don't cut studs, just
position the registers in between. For privacy, the louvers should
provide enough. If sound transmission is a concern, you might want to
vertically offset the vents on either side and use the wall cavity as a
short duct, lining the interior of the drywall (between your two cuts)
with a duct liner such as :
http://www.knaufinsulation.co.uk/out.../page_394.html to keep
the sound from bouncing between.

I don't think you have to create a frame inside your cut outs since
each stud cavity is likely sealed off. It certainly wouldn't hurt, but
is probably more work than is necessary.

I very much dislike doors that are undercut. For one thing, hot air
rises and an undercut door is sucking cool air out and leaving the hot
air in the room. Besides, it just looks bad.