Room correction help

First best thing you can do is forget everything you think and pretend you know nothing and neither does anyone else. Because its true. The only way to know is to find out. The way to find out is to try and see.

So get yourself a panel or two of Owens Corning 703 acoustic panel. OC703 is cheap and handy. OC703 is what's inside a lot of expensive acoustic panels. Many hardware stores will have it. Its light weight and can be easily cut to whatever size or shape you want with nothing more than a razor blade or box cutter.

So you want to know what the rug will do? Lay your 2 sheets of OC703 on the floor where the rug will go. Listen and see. Takes like 2 minutes and you will know.

First reflection? How much difference will it make? Lean one panel against the wall in the right places and again in a few minutes no more guessing, you will know for sure. You can even experiment, by say cutting a piece 12" square or whatever and sticking that on the wall with tape or push pins. The stuff is really light, only takes a few minutes and you will know.

Keep on going. In no time flat instead of hoping and dreaming someone here knows, forget about it, you will KNOW what works, exactly, and by how much, and which was better and which was not as good and which was a total waste.

I can tell you right now the most effective locations, which no one else will, because they haven't bothered to try and find out- cut triangles about 12" on a side, stick them in the corners where the ceiling and walls come together. Next most effective, 8-12" wide rectangles in the corners where the walls meet, and then after that where the walls meet the ceiling. 

Putting it on the side walls works, but its real easy to over-do it and wind up with too much damping.

But the real reason for taking this approach is you learn to listen and evaluate. Then you order up a full set of Synergistic Research HFT for your walls and speakers. These tiny unobtrusive little things are about a million times more effective than anything else you can do. But to fully appreciate just how good they are you kind of have to work at it the old school way first.

@millercarbon

Next most effective, 8-12" wide rectangles in the corners where the walls meet

You mean covering the corners like corner traps, the exposed face being 12" wide?

FWIW I experimented on first reflection points and I didn't hear much difference. Maybe these spots you mention male a bigger difference. I'll try them out. I really wanted to hear a difference, but didn't. 

A couple of things to think about:
normally you might want to place your speakers in the right location before applying reflection point panels. There is a rule of thirds or maybe 5ths meaning your should start your speakers 8’ out from the rear wall. This should help in bass response and depth. 
The space between your speakers seems it might be too large. This will hurt your soundstage and imaging. 
Check out the Cardas website along with many others that give you formulas On where to place your speakers based on your room dimensions.

I'll vote with cal18, although using an older Behringer 8024 vs. the newer 2496.  I've been doing this for...damn, nearly 3 decades, starting with an AudioControl C101 long ago and far away....

Never been one to clutter up a room with panels, traps, etc.  SAF is one concern, but her primary concern is being able to use the equipment without a degree in audio engineering. ;)

Rugs, curtains, and some 'tasteful additions' can go a long way.  All homes I've lived in have always been problematic acoustically.  The only way one can achieve 'Perfection' is to have the space 'built for purpose' which most of us can't achieve....

As Clint noted: "A man has to know his limitations."

The Answer: 2 systems.
-Main living space set up for Simplicity, but properly eq'd and capable of pleasing the both of us.
-My 'play space', which is just as insane as I might care to have it. *G*  (But capable of sending a feed to the mains in the living room...have your cake, and listen to it, too....*smirk*).

"Thinking outside the box" is easy.  Stop thinking there is a box to be outside of.... ;)

Diffusers and general room treatment are great, but I think you need to address speaker placement first; it's cheap & doesn't affect WAF. At 2.5ft from front & sidewalls may not be optimum. I recommend you try 2.5-2.8' from side walls & ~4.5 from the front wall and adjust thereafter for the best bass. Then place the flokati in front in-between the spkrs & sitting position -- or wherever the wife prefers (if it kills the sound she will hear it & correct).