Hi there Aaron, and Welcome!
so I need to figure out a way to isolate the room some if possible.
As Francis already mentioned, unless you can define how much isolation you need, in decibels, it's pretty hard to help you design it! That's sort of like you saying: "I'm here, and I want to go somewhere", but not saying where "somewhere" is! Is it North? South? Far away? Nearby? Would you need to take a jet to get there, or would a bicycle be sufficient? That's the dilemma you are bringing.
1. Unsure of actual dbs of loudness,
Continuing my above analogy, this answer is equivalent to saying "Unsure of actual distance to 'somewhere': It's just sort of maybe roughly that far, I think". Not much use.
but I do like to mix loud sometimes. ... This will mainly be used for mixing and recording vocals/acoustic guitars.
... continued analogy ... "I think 'somewhere' is mostly far away, sometimes... but I want to get there in a short distance, too"...
but I would like advice on cheaper ways of getting at least a little less isolation from the bedroom next door and the back wall to the closet
analogy: "I would like advice on cheap ways to travel to 'somewhere' that gets me a bit closer to it, in the right general direction.... but I don't know what that direction is either. It's just 'somewhere. Do you think I should pack a bag?" ...
If I did a room inside of a room design then I would lose precious real estate...
..."If I got on a plane, that would cost a lot of money, but I don't even know yet if a plane is what I need, or maybe a skateboard, because I still don't know where 'somewhere' is"!
To increase isolation: I'm thinking of...
Analogy: "To get on the road to 'somewhere': I'm thinking of . . . doing stuff that might take me to somewhere, or it might take me in the exact wrong direction, and it might be too far, or it might not be far enough... I really don't know... but at least I'd be traveling! Even though I don't know if it would be towards 'somewhere' or not... "
OK, enough of that analogy: The point is you need to define where 'somewhere' is. It is one thing to build a room for 40 dB of isolation, and a very different thing to build it for 60 dB isolation. In fact, it is very literally 100 times different: You need to block
one hundred times more sound to get 60 dB of isolation, as compared to 40 dB... even though subjectively it would sound only one quarter as loud...
So get yourself a sound level meter and measure the level you normally mix at, when you have it turned up high! And while that is playing, go out to other rooms around you, and measure how loud it is in there: the difference between the two tells you how much isolation you are getting at present. Then turn it all off, and measure the ambient level in the building late at night when it's very quiet. The difference between that measurement and the loudest one, is how much isolation you need. It's not hard to do! A hundred bucks will buy a decent meter, and a few hours will be more than enough to get the readings. Use "C" weighting and "Slow" response on the meter.
Based on that, we can show you different methods of construction and materials that will provide the level you need. Then you can decide if you are prepared to do it or not, in terms of the investment in time and money, and the amount of space you would lose.
For acoustic treatment: I was thinking of taking the doors off the closet and using it as a giant bass trap? Seal the inside drywall, Fill it with insulation and put a fabric layer across the front? I would need to do some type of bass traps in the corners, wall dampening and a cloud...etc
That would be a start, yes. But you'd also have to set up your speakers and mix position in the correct locations and relationship for that room, and add the treatment at the first reflection points.
which wall would best help the accuracy of the speakers?
None of them!

No wall will help a speaker to be more accurate. In fact, the exact opposite! Having a speaker in proximity to a wall will REDUCE the accuracy of the speaker: It will distort the power balance, as well as creating a whole series of undesirable artifacts, such as SBIR, comb filtering, reflections, etc. That's a simple consequence of having a speaker anywhere near a wall, floor or ceiling. So NONE of your walls will make your speakers more accurate. What you do need to do, then, is to put the speakers in the spot that will create the least distortion artifacts, then provide the acoustic treatment that will help to control some of those artifacts. The best way of doing that is to flush-mount the speakers in angled sections of the wall that you would build out, called "soffits". That might not be an option for you, but it's the single best thing you can do to your room and speakers, acoustically, for a simple reason: it removes the front wall! If the speaker is flush with the wall, then there are zero artifacts: the speaker does not "see" the wall any more, and the wall (or more correctly, the "soffit baffle") prevents those artifacts from forming. There will still be other issues, caused by the side walls, floor, ceiling and rear wall of the room, but at leas the "biggie" is out of the way: no front wall.
If you can't (or don't want to) flush-mount your speakers, then the next best thing is to place them tight up against the front wall, where the artifacts will be mostly in the mid-range of the musical spectrum, and that's a place that can be treated. If you have the speakers away from the front wall, then those problems all move down to a lower frequency, where they cannot be treated.
If the closet was turned into a giant trap...would it be better for the speaker to be facing it or away from it?
Towards it. Always. You want your speakers wet up against a hard, solid, rigid massive surface, (with suitable treatment) and facing down the longest axis of the room, towards something that is absorptive (and maybe also diffusive). The reason is simple: the room must not "color" the sound that you hear at the mix position. The room must be set up and treated such that you can ONLY hear the direct sound coming from the speakers. After each sound passes your head, it must not get back to your ears (by bouncing off other surfaces and returning) until at least 20ms has passed, and during that time any reflections that do arrive should be at least -30dB quieter. After the 20ms is up, some reflections can start arriving at about -20dB, but they must be diffuse, and they all of the reflections after that must die away slowly, at a constant rate for all frequencies, until you can¿t hear them any more. The total time until the "point where you can't hear them any more" is called the "decay time", and there is an equation for calculating what that should be for your specific room. At a rough guess, it will likely be around 200ms, but you'd have to do the math to be sure.
So, you need enough treatment to ensure that the sound that goes past your head will meet those specs: be attenuated and diffused by the rear wall, and arrive back at your ears after the correct delay, at the correct level. That will require deep, thick absorption on the rear wall, so filling your closet with suitable treatment would be a good idea. That treatment needs to be tuned such that it does not absorb too much of any frequency range: just the right amount, so that all frequencies die away at the correct rate.
There's more to it than that, but this should at least get you started.
but as I'm sure you can understand they are very busy with other things so don't worry if you don't get a response from them right away, it may take a few days

We are only human! And we have lives outside of the forum...
The reason why they ask how loud you are and how quiet you need to be in dbs is because it will help set your isolation expectations,

And it also defines the type of materials and methods that will be needed to get there. A sheet of cardboard will give you 10 dB of isolation, but if you want 80 dB then you are going to need massively thick double, reinforced concrete walls, floated on steel springs! One of those options will cost you a few cents, the other will cost you a few hundred thousand dollars, at least... (In other words, defining your isolation needs in decibels, will tell you exactly how far 'somewhere' is, and which direction you need to go to get there!

)
Doors and windows are often the weakest link in studios so if you also have a window, which I think you do since it's a bedroom you'd probably have to research what options you have to address this issue.
I'd scrap the carpet as believe it or not they are bad for studios.
Thanks for chiming in, Francis! Your comments are spot on!
- Stuart -