[T-Mobile, Samsung Galaxy SIII] shrimants:
You seem to be in approximately in the same boat as me, or in any case you will be in the same boat as me given a couple days worth of idle research. Hopefully you subscribed to the thread so you'll get this as an email reply.

Currently, there are a few roms: Cyanogenmod, AOKP, Paranoid Android, Stock, and variations of these 4. Cyanogenmod is as close to AOSP (google's definition of android) specifications as you can get. AOKP is based on cyanogenmod, but it has some other tweaks thrown in to make it look different and cool and whatnot. Paranoid Android is also based on cyanogenmod, but it is scalable. IE you can tell your apps how big/small you want them to be on your screen and put them into "tablet mode". The variations are the exact same thing, except they all have the specific users' own spin. Lets say I liked Cyanogenmod but I wanted a specific kernel and a couple of optimizations. I'd put those in, make it a ROM, and now that is my own "variation" of cyanogenmod, based on cyanogenmod.

Please understand that this is an extremely simplistic explanation of everything. EXTREMELY. you can go to the respective projects' chat rooms/forums and ask questions about them there, and sometimes even a lead developer will answer you.

Next is your question about getting roms. According to the android "licensing" or whatever you want to call it, companies who use android as their operating system are required to release a certain amount of source code for their device. I think what this means is that for Samsung, since they are currently officially on ICS, they have to release the ICS kernel. Sony is extremely nice and releases a butt-ton of information. The Nexus devices are google devices and adhere 100% to specifications, and thus are always ahead of the game when it comes to roms and such. For the GS3, we have to figure out what Samsung did to make their publically released source code run on their hardware. There are 2 ways to do this: you can either hack around and reverse engineer all of their hardware/drivers or you can create your own custom drivers for everything to run. If you have any experience with linux, you should know what a daunting task this is. Imagine plugging in your webcam to your computer and nothing happens. You know your computer can see it, but now you have to write the software for your computer to work with the webcam with little to no documentation provided by the webcam maker or the computer maker.

So, Right now no one "has roms" for the device. I'm not entirely sure how they managed to get jelly bean running on the GS3, but that is why nothing is definitely stable yet. Once samsung releases their code, they will be able to cross reference and see how they are doing things and fix a lot of issues, but for now everyone is approximately on the same page. There are minor bug fixes that happen here and there but nothing major will be happening any time soon.

There are a couple ways you can figure out the status of a ROM. you can either read the changelog (quick, not very helpful) or you can scan through the entire thread and read that day's activities (slow, very helpful).

For a quick tip, Cyanogenmod Build 25 (september 25th nightly) seems to be very stable. If you want to give any rom a shot and dont know which to try, CM is the way to go, and Build 25 is probably your best bet.

If stability or "daily driver" is a key issue for you, though, it is best to remain stock and deal with the gentle oversights of the stock rom and have it pretty stable than it is to go to cyanogenmod and instantly start tweaking everything.