New comer looking for help to buy!!

I had a 5.1 channel setup when I had the sofa back against the wall. And the setup was great for movies and gaming. I was able to hide flat speaker cables under the carpet. Well yeah I was gaming back then!!

I remeber paying like $3.5k for paradigm studio 5 channel setup. Except for sub all were used back then.

If you want to get something for $2k try used market.

Btw not sure if you watch 4K content but depending on your tv size 10 feet may be too far. You may already know that , thought I should mention.

Boston Acoustics A-series speakers are well regarded ones on sale now at Accessories4less.com   Entire 5.1 set is $800.

If you don't need 4k feature in the receiver, B&K 7.1 receiver off ebay used sounds real good - better than the mass produced Japanese offerings.

Yes I would definitely go with a 5.1 system since you have un-closeable openings out to other rooms.  This I would say is your most significant challenge to reproducing great audio. You need to "fill" your room with  sound.  Your listening room should have as much upholstered furniture, drapes and carpet as possible to help deaden the liveliness of the environment.  I have the same situation at home took these steps.  Now as far as your budget goes, $2K may be a bit light to build a brand new 5.1 system so you may want to consider purchasing well cared for pre-owned components on Audiogon.  Make sure the seller has a high rating.  How about this: a 5.1 AVR (i.e. Marantz, Yamaha, Onkyo, Denon, NAD. etc), pair of decent bookshelf loudspeakers (i.e.  B&W, Monitor Audio, PSB, Boston Acoustics, etc), CD player (Marantz, NAD, Sony, Denon, etc), pair of surround satellite speakers, powered sub-woofer and cable.  If you browse deep enough in the Audiogon equipment for sale link, I'm sure you'll find a great combination of pre-owned equipment.  I would purchase preowned over brand new mass market components.  Just my opinion.        

A really good room correction system can fix a lot of ills.  I just bought a5.1 AVR from Anthem that will be installed on Wednesday and am itching at the bit to hear.
DIrac is the other well regarded system.  My advice is to get whatever speakers you want and then an AVR or processor with good correction.

Anthem installed today...major sonic upgrade

Went down that rabbit hole .Marantz processor went .4k to get back in the game . Downsizing back to 2 channels.