Recess AV Cabinet in Bonus Room
Here's what I'd do:


Cut an opening in the drywall appropriate for your cabinet. Allow for the thickness of your cabinet walls. Cut off the one or two studs that are likely to interfere with the opening, from the back side, 1-1/2" above the opening in the drywall. Cut out the bottom wall plate flush with the sides of the rough opening.
From behind, frame the sides and head with single 2x4s. If desired, add a flat 2x4 on the back side of the wall, spanning the head of the opening, to stiffen the head area. It should extend at least to the adjacent studs and be flush on the bottom with the opening. 
Add any floor sheathing needed to support your cabinet. This can be roughly fit and somewhat oversized. 
Set your cabinet and trim it out to the drywall as desired. 
From the back side, apply vapor barrier and insulation in accordance with your home's existing situation. This could be batts simply suspended from above and tacked at the bottom by nails or whatever.


There's no need to frame up walls around your cabinet unless 1) they'll form the cabinet walls, as in a drywall compartment, or 2) you need them to support insulation.