I don't see any issue with the double door. Its better for bass response if you leave it open. You will likely have to mount acoustic treatments to the doors.

Your going to need alot of bass trapping. The ceiling cloud can be thick enough to tackle some low room modes without interfering with height too much.

Id plan on covering the entire back wall with bass trapping and possibly broadband treatment in front. Then hit all the corners.

Typucal fluffy fiberglass is the besr bang for the buck for bass trapping. Rockboard 40, and Safe N' Sound Insulation are excellent for broadband panels.

You may consider something on the rear wall that has slats (tuned to the room mode) in the top and bottom 3rds and is porous in the middle. This gets you trapping without sending high amplitude reflections back into the mix spot.

You could also angle the panels on the side walls and make them (semi) reflective. This way you direct sound away from the mix spot but don't kill all the "life". Maybe use this approach for the rear half of the room, with pourous broadband absorption in the front. You could do something like a 1,2,4 sequence for the slats, with the spaces between them being 20% the width of the board. This gives you dispersion over 3 octaves and bass trapping below that. They are essentially highpass filters letting only the lows into the trap. They will reflect sound towards the rear wall.

I suggest you get the free REW room measurement software. Its also got some room simulation capabilities. You should test the room before treating it since the walls will leak sound, it can skew which frequencies need attention vs the calculations (which assume lossless walls). This works to your favor since the walls will let out some bass acting as a bass trap.

https://amcoustics.com/ has a great set of simple free calculators/tools.

You can use acousticmodelling.com to see the absorbsion characteristics of various types of traps/absorbers and how dimensions and material choice effects their performance.

Rod Gervias build it like the pro's has drawings of how to build various types of treatment, and is a generally awesome resource for anyone involved with a studio.

I will post a couple links about finding the best speaker and listening spot, and one that shows the basic idea of the treatment. Eventually i will write up my own post outlining these processes so im not re-directing people to GS, but (the late) Boggy was a great designer so his words should be seen.

Boggy on speaker location

https://www.gearslutz.com/board/bass-traps-acoustic-panels-foam-etc/744554-speaker-placement-methods.html

How to find the listen position

https://www.gearslutz.com/board/showpost.php?p=7377799&postcount=6

Jens outlines basics of treatment including pictures, with links about resonators

https://www.gearslutz.com/board/showpost.php?p=14291770&postcount=1

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Without rebuilding it, you're not going to keep loud sounds in or out. But you can do a fair bit of treatment so you can have a better chance of trusting what you hear. The nearly square footprint works against you because it will tend to favor a specific resonant frequency range. I suspect general bass trapping, with an emphasis on first reflections will be the first thing to look at.

Hi Paul, welcome to our community. Others here are much better with answers to your acoustic questions but while waiting, here is a ton of links that may have some answers.
https://recording.org/tags/room-acoustics/