-Do these wall art panels need to be a certain thickness to be effective at sound absorbtion?
A sound panel does not really need to be thicker than 2". You only need really thick panels if you want to treat/absorb bass frequencies. I don’t think you will have bass node issues in your very large room and connected rooms.
It makes sense to put the speakers wider so that they are in front of the windows for the window treatments to absorb the sound. Will look into acoustic panels, larger rug and artwork on the side wall. The back wall is going to be an issue
There is a reason to put absorption right behind the speakers. When sound is generated by speakers, the sound waves will be reflected by the walls behind your listening position. Many reflections make their way back torwards the speaker and are actually reflected again by the wall "directly behind the speaker". This double-echo is what you want to avoid. That is the primary reason for recommending you move the speakers wider to be in front of the windows and use very heavy curtains as a sound absorber for mid and high frequencies.
Ultimately, that whole tile floor as well as the tile wall behind the tv is what is killing you. Drywall is pretty okay for acoustics, but you still want to catch first reflections.
Wall behind TV - I would look into maybe doing a heavy curtains here. One heavy curtain behind the tv to cover the tile - or possibly a diffusion panel to cover the tile. The diffusion panel could get expensive. Then heavy curtains for the windows. If you use curtains for both tile and windows, maybe think about floor-to-ceiling curtains so that the window/tile sections all look uniform (this is good for wife-acceptance-factor).
Left wall - you currently have this multi-colored painting. I would look into getting a couple of GIK Acoustics art deco panels. Maybe a couple of 24x24 panels. Or even a big 48x36 if you want. You don’t need a whole lot of treatment on this wall, but you do need some. The panel should come down to the level of the couch. Good hint - put a mirror on the left wall and move it around. If there is a point where you can see the speakers from your listening position on the couch, you definitely need a panel there. See here:
https://www.gikacoustics.com/product-category/acoustic-art-panels/
Floors - put as many large thick area rugs as possible. You have a small rug in front of your couch and I also see a small rug under that coffee table. I think the tile floor right in front of the speaker should get thick rugs for sure.
Backyard windows - this is likely far away enough where it won’t affect acoustics too much, but you can put curtains on these windows as well if you still need a little bit more echo absorption. This will help.