The world is your oyster in that room given its unfinished state.
Some thoughts:
Flat panel tv versus projector - If you want to keep the budget down, a large TV can be had for a reasonable price. And TVs are nice because the image from a current-crop tv is fantastic---incredible sharpness, brightness, deep blacks, great contrast. And---they're silent---no fans.
However, there's just something about a projector presenting an image on a quality screen that feels like you've been transported to another place. I love my [now 3 year-old] JVC projector and Stewart projection screen, but there's no doubt---it's not as crisp as a tv panel. But movies in theaters usually have a bit of a diffuse look, so you have to decide what you're going for.
Other things to consider with a projector:
-Dark scenes can be a challenge for a cheap projector on a big projection screen.
-The projector hangs from the ceiling above the seating and you can hear the cooling fan. And when you play 4k content like a 4k Apple TV, the fan kicks to a higher speed versus 1080 where it's pretty quiet. And if you live at altitude (I live at 6600'), the fan is supposed to be setup to run even faster due to thinner air, although I don't it since the projector is not in an enclosed space.
Anyway, with the projector, I kick back in a reclining chair and I'm excited for theatre-style visuals. And the setup delivers. But, there's no doubt, a good projector and screen definitely adds quite a bit of money to the project over a tv.
Speakers - These days, there are plenty of good speaker options without spending a fortune. And because you have a raw space to start with, planning for speakers and wires should not be a problem. I'm a fan of the Dolby Atmos format. Aside from the standard 5.1 or 7.1 surround setup, if you can put several (not expensive) Atmos speakers on the ceiling, it's REALLY immersive. Stuff is literally flying over your head. Fantastic. I would go for more speakers that are cost effective than more expensive speakers but less of them.
Make sure you have access to wires behind walls/ceilings for future service. There was an existing setup in the house when we bought it and we had to upgrade the HDMI cable for the new projector/receiver setup and it was a NIGHTMARE running the new cable. Had to bring in professionals who had a heck of a time running the new cable above the ceiling.
Acoustic treatment - Before you think about acoustic treatment, you will have to decide how much you are finishing the unfinished room. I can't exactly tell how much the wood support post encroaches into the room, but I might consider a straight wall that goes across the whole front of the room to eliminate the bump-out in the front. This will allow you to center your screen in the room versus pushing everything to the left. And you could plan to run wires behind the wall and keep things clean looking. And in that front wall, you could put studs or plywood exactly where needed to support heavy stuff.
Let's say you finish the whole space with dry wall, an 11'x14' room with parallel surfaces and a fairly low ceiling is a recipe for muddy/washy acoustics. (I do music for a living. And I used to mix television shows for a living. I've put my time in sound/music studios.) In a room like that, I would recommend acoustically treating the walls somehow (and it would benefit you to put some stuff on the ceiling too). People never get excited about spending money on foam panels to reduce sound reflections. But you gotta do something if you want to avoid muddy sound. I've been a fan of Auralex foam panels over the years for my music work. They're simple to use and they've got a bunch of shapes, thicknesses and colors.
If you don't finish the walls, you can do more vibey things like putting full length floor-to-ceiling curtains on the sides and back of the room. They do an incredibly good job of killing unwanted sound reflections. And curtains can look like anything. They can give a cool funky look to a room. And of course, carpet helps with unwanted sound reflections. I've been in some incredibly high end music studios that have curtains and swooping fabrics coming down from the ceilings to kill reflections. The studios are super vibey.
Receiver - get something that does 4k, Atmos, has a bunch of speaker outputs and has a bunch of video inputs for all your different viewing devices---Apple TV, cable/satellite box, game console, (BluRay player if that's even a thing anymore....), etc. Most anything modern should be fine. I've had good luck with Denon.
Universal Remote - I've always been a fan of some kind of Universal remote--especially for non-technical family members. If your electronic hardware is behind you or tucked away, then an RF remote is probably the ticket. An IR remote that has the repeaters for hidden components is ok.....but....I think RF is more reliable.
Seating - I am a personal fan of reclining chairs. There are lots of options for a three person couch where two or maybe all three seats recline. And they don't have to cost an arm and a leg. We got stuff that power reclined with lighted cup holders and USB chargers that was reasonably priced. We have a mix of reclining couches and reclining individual chairs. Works great. The individual chairs make up the first row. They're placed wide on the sides and actually sort of encroach in front of the "main" couch that is the second row. But it totally works. The individual chairs do not get in the way of the view of the second row. I built a platform for the third row couch so it sits higher. We can seat 7 efficiently in an area that's not that long.
In my situation, the house came with a movie room area that was nicely built out, but all the equipment was 12+ years old. So, I replaced many components. And we put our own seating in. The previous owner had installed 4 speakers in the ceiling as surround, but I changed the whole audio setup so the ceiling speakers became Atmos and I put some small speakers in the back for surround. It's so much fun to watch movies in there---dual subs, Atmos surround, projector, reclining seats. My theatre is not a fully enclosed room just by itself. There's no left wall. There's a bar to the left of theatre and we keep it stocked with drinks and candy and microwaveable popcorn. Since it's a wider open space with many articulating walls and recessed ceiling features, fabric on some of the walls, a ton of carpeting and the chairs are soft fabric instead of leather, I didn't feel the need to put any sound additional treatment on the walls.
Here's how it looks:
Have fun with your project!