For the projector, the Panasonic PT-AE8000U would be a good choice. At 22 feet, it'll project on any size screen from 112" diagonal (98" wide) to the limit of horizontal width of your screen wall.
Play with this projection calculator. We'll estimate $400 for a DIY screen (material from
Carl's Place + wood frame + L brackets and screws from Lowe's + black velvet tape border + a string of $20 LED lights to line the back edge if you feel like it). *shrug* That'll leave you around $2800 for speakers and AVR.
I recommend jumping on three of
these JBL Studio 580 for your front LCR as soon as you can. No telling how long they'll be that cheap ($389 each as of this post). They're usually
twice that much or more, and the JBL store has already reverted their price back to full retail. If you can score them, that leaves around $1400, but you'll have to source some acoustically transparent screen material from somewhere other than Carl's Place. I don't see any on their website at the moment. One of the other veterans can probably recommend somewhere.
Subwoofer:
Hsu VTF-2 MK5. That'll leave about $800.
AVR: something cheap.
Yamaha 5810 refurb, leaving ~$500.
Side in-wall surrounds:
Monoprice w/ AMT tweets, leaving ~$400.
Pair of in-ceiling speakers:
Monoprice w/ angled drivers, leaving ~$320.
My poor math: probably $320, leaving you with your entire budget spent. Enjoy! Spend the rest on Monoprice CL2 speaker wire, a Monoprice subwoofer patch cable, maybe consider upgrading the sub to a Hsu VTF-3 MK5 or adding acoustic panels, or budget for a Harmony remote. Or the acoustically transparent screen might eat up the remainder of your budget. Don't know.
But if you want more speakers, you'll have to increase your budget or go with a lesser quality front stage. 9.x channel Atmos-capable AVRs jump up an additional $400 on Accessories4Less I believe, then the speakers themselves will add another $80+.