I looked at your tool list and have the following comments;
Use Robertson screws whenever possible, better torque than Philips.
Don't use window screen to cover your panels, use fabric or speaker fabric, both at Fabricville and affordable.
Don't glue the fabric, in time it will fail, staples are the better option, in 3/4 MDF you will need to hammer them in for they usually don't go in all the way.
The design shows only 3'' deep, not enough, the soundproofing must fit only snugly not tight to be effective, it is made to fit a 2 x 4 which is actually 3 1/2'' deep, so go to that or even 4''.
You should screw into the studs, not drywall anchors, these boxes will be slightly heavy. You could also use the French cleat system you were considering.
Don't fear your tools, pay attention and respect the tools and avoid injury.