Looking at your room layout, I'd close off that section you mentioned and make it a kids' area. Maybe for fun install a dutch door?
As far as where to put your screen...
On the one hand, if you put it against the far right wall, you could get in two rows of seating, but then depending on the size of your screen and if you chose not to go with acoustically transparent, you may have a tough time fitting (I'm assuming) floor standing speakers to the left and right of your screen. Also, from an acoustical standpoint you definitely don't want your first row of seating to be midpoint in the room. Ideally IIRC you want your main listening position to be about 1/3 from either the back wall or the front wall. That would make your first row of seating around 13 feet back from the screen. For some, that's too far away. Answer? Go as big as you can for your screen, and go with an acoustically transparent screen and put identical speakers at ear level across the front, left and center. From my experience, having the same speakers at ear level behind an AT screen has been revelatory. Also, you'd have lots of room to install ceiling speakers for Dolby Atmos if you chose to go that way.
Your other option I suppose would be put your screen against the long wall opposite the would be kids play room. You won't be able to go with a massive screen because you'll be sitting closer, but you could easily get four or more curved or straight home theater chairs in there. You wouldn't need to go acoustically transparent as you would have lots of room to put your L/R speakers. Also, since there's lots of room to the right and left of your screen, that will allow your L/R speakers to really open up. In the other config, you'd definitely want to put absorption panels at the first reflection point for your L/R speakers. Projector wise you're also limited to one that has a really good zoom/throw ratio (this config may rule out a DLP projector). Sorry if I'm inundating you with things to think about, but these are things that I've sorted out over the last few years.
If it were me, I'd go for the first option. Do lots of reading and research before you jump in. Be lazy, do things right the first time.