Building a New Wall with Projector Niche

I’m hoping to transform my basement into a home theater space.

While there’s a lot involved in the project, the primary scope involves (see pictures below):

1. Close the opening on the front wall
2. Create 100” projector screen niche (recessed rectangle) on front wall
3. Demo and dispose of popcorn ceiling, install new ceiling drywall

To create the projector niche, I can’t go “backwards” (behind the existing drywall) because there’s foundation right behind the wall studs. I need to create a new wall that is 6” forward (from existing) in order to have the space to frame the niche. I say 6” because I’ll have a soundbar that is 4.5” deep that will sit on the bottom of the niche, so I’m planning using 2×6 framing lumber.

So, with that in mind, my primary questions are as follows:

1. Should I remove the existing drywall before creating the new wall that’s extended forward? Or just build “on top” of the existing wall? What implications does either choice have for closing the current wall opening? Note that there’s two electrical outlets in the wall opening that I’ll want to bring forward.
2. What’s the best way to secure the new beams to the old beams? And the wall framing into the ceramic tile below?Construction adhesive?

As an aside (as it may impact answers to questions above), while I’m doing this work, I figure I might as well remove the popcorn ceiling. While I’d normally wet and scrape, I can see seams showing that would probably appear as cracks without the “popcorn”. So I’m planning on taking the ceiling out and installing new drywall reinforced with liquid nail.

I’m also on the fence about extending the existing ceiling “bulkheads” to create a coffered ceiling (or ceiling soffit?), as it may add to the home theater aesthetic…

Finally, what’s a versatile saw for accomplishing the tasks above? Miter? Circular? Table saw?

Thanks!