I"m building a frame behind the wall & above the fireplace to insulate a bare wall. Tips please.?

I have this fireplace wall facing the north wind, & I discovered that it's just drywall all the way up to a 15 foot ceiling. No insulation behind the drywall at all. No wonder the heater has to turn on every 5 minutes.

I plan on building a 2x4 frame with 16" studs on center. Then stapling faced R-15 insulation on the drywall side. Then using mdx or whatever that plywood stuff is to seal it in place so there's absolutely no chance of it ever falling down on the fireplace insert.

I noticed Home Despot has 3 different types of 2x4's. Which one is the hardest type, & do I need batting in between the 2x4's already there & this new frame or is it common practice to just nail 2x4's together such as in door frames.

I noticed home despot has 3 different types of 2x4's.