Building a New Wall with Projector Niche
This looks like a fun project.  I want to start by linking a few great books I own and like.  They're well-illustrated and you can find them in many home improvement stores &amp; libraries.  These will really help you a lot since you are new to framing and maybe electrical as well.

Codes for Homeowners
The Complete Guide to Walls &amp; Ceilings
The Complete Guide to Finishing Basements
The Complete Guide to Wiring

Existing wall demo?
With that said, personally, I would demo the existing wall rather than leave all that drywall &amp; electrical inaccessible.  I'm not aware of any requirement to remove the old wall; it's just what I would do.
Framing lumber choice
You mentioned using 2x6 framing lumber to build the new wall.  That is unnecessary; you can use 2x4.  Your new wall should be secured to the &quot;ceiling&quot; (really, joists or blocking above the ceiling) and the floor.  It doesn't necessarily need to be in contact with the foundation wall or the existing finished drywall (even if you don't demo it.)  Therefore, the extra depth of 2x6 lumber is not beneficial.
You ask, What’s the best way to secure the new beams to the old beams?. Both the &quot;Basements&quot; and &quot;Walls &amp; Ceilings&quot; books cover this with illustrations.  But basically, there should be joists above the existing ceiling, and you want your new wall to be secured to them.  If they run parallel to the new wall you might add some &quot;blocking&quot; between two joists and then nail your new wall to the blocking.  The pictures in those books will help a ton.
Saw choice
A circular saw is a good choice and a great &quot;first saw&quot; for someone getting into serious DIY projects like this one.
You won't need a table saw for this.  A miter saw might be helpful for a few cuts when you're ready to install new baseboards, chair rails, or other trim; but don't buy a huge 12&quot; miter saw; consider renting one unless you're sure you will use it again.  They take up a lot of space.