First, high ISO performance.... well, you don't mention what you've tried, so it's hard to make recommendations. All the current Canon APS-C cameras (except for the T3/1100D) have essentially the same sensor and high ISO performance. Each of us has out opinion of what's acceptible. I'll use my 7Ds to 3200 without too much concern, though I'd prefer to keep to 1600 or lower if possible. Higher than that, maybe for black and white conversions... but not usable for most of my purposes.
You might find different levels of high ISO noise acceptible. Or you might benefit from using different software to reduce noise in your images. (It appears the latest Canon FF models are getting most of their high ISO gains from the image processing, not from significant changes to the sensor.)
No, it won't be "massive" improvement to go to full frame. I also use a 5DII and IMO it's good for about one stop higher ISO images than my 7Ds. I'll use 6400 with the FF camera, vs 3200 with the croppers. Looking at 5DIII and 6D, shooting RAW they don't appear to offer a lot higher, less than a stop's worth.... but they do seem to give one or two stops higher usable ISO when shooting JPEGs.... Again, it appears to be in the post-processing where the high ISO noise reduction is taking place.
That tells me it should be possible to accomplish more with NR softwares with any of these cameras, perhaps Canon's own DPP in the latest version, perhaps with third party image processing software and/or NR plug ins. I saw a major improvement in noise reduction going from Lightroom 2 to LR3 and from Photoshop CS4 to CS5. I probably could accomplish more with additional and newer softwares. Look up some of Teamspeed's high ISO work with 7D here on POTN. He's taken it to the limit and gotten really impressive results. Keep in mind that the 7D has essentially the same sensor, processor and high ISO potential as your T2i.
One thing that's critical with any DSLR to minimize noise, you have to get the exposure correct. If you have to increase exposure, lighten up the image at all in post processing, you will increase any noise in the image. It's better to be slightly overexposed and pulling down exposure or darkening the image in post processing. But you have to be careful, emphasis on "slight" overexposure... too much and there will be unrecoverably loss of detail in the image's highlights.
Second, focusing. There actually are three major factors effecting focus performance. The camera is only one third of the equation. Your camera is pretty basic, but still capable if used relatively simply. I'll presume that you know to use AI Servo for moving subjects. On T2i only the center AF point is the dual-axis or "cross" type, so for best performance with AI Servo you should limit to using only that more sensitive and responsive point. This is often the best method with all the cameras mentioned, no matter how points they have or how sophisticated their AF systems. The T3i/600D and 5DII have similar AF point array to your T2i/550D (though the 5DII has some additional "hidden" AF points that only work in AI Servo mode, when optionally enabled). The 60D and T4i/650D have an enhanced version, all nine points are dual-axis, cross type. 6D has 11 points, with only the center one cross type. The 7D has 19 points, all cross type. 5DIII has 61 points, with 41 of them the cross type. In all cases, the camera's center point (or in the case of the 5DIII, several points near the center) are enhanced and higher performance... even the ones with all or many cross type points throughout the array.
More is not always better, though. It might give you the ability to place the focus point more precisely, but it also means more to manage while shooting. The processor is another factor. Most of the cameras use the same chip that's handling images to control AF. The 5DIII, T4i and 6D use a newer Digic V processor that's higher performance, while the rest rely on Digic 4. The 7D uses a discrete chip for AF, and has separate dual DIgic 4 for image processing.
The second major factor is the lens. A USM lens (or equivalent from third party such as Sigma HSM or Tamron USD) is in most cases faster and more accurate focusing than lenses with micro motor focus drive systems. If your lenses are not USM, you aren't going to see much improvement going to a camera with a more sophisticated focusing system. Also, a lens with a larger aperture (f2.8 or larger) delivers more light to the AF system in the camera, that helps it perform better.
The final major factor effecting focus is the user. Ultimately, the ball in in your court, to do your job as the photographer to select a good target and keep an AF point on the subject, maintaining focus. Don't expect to get 100% perfect focus all the time with moving subjects. You'll miss a few... we all do. But with practice and familiarity with your camera and lenses, you will find you get better and better results over time. Practice pays dividends. In combination with AI Servo, you also might want to experiment with Back Button Focusing, which is a technique popular with sports photographers, but also is very effective with any other type of moving subject. It's also usable with stationary subject and can be used with One Shot. A lot of people who learn it wonder why they ever would use any other method of focus. Even though they call it "auto focus", there is still no substitute for a good, skillful user behind the camera. You also might take some time to view this video, the first of a series of three on Youtube, all about using the Canon autofocus system. Even though it's a few years old now, much of it is applicable to your camera. You'll want to watch all three. The latter part of the last video in this series might give you some more idea what some of the other cameras, including ones you asked about, can do.
Finally, going from a crop camera to a full frame camera is more than just buying a different camera model. Chances are, you will also have to invest in a different set of lenses, though you didn't mention what lenses you have now. There is a smaller selection of lenses for full frame (only EF or EF equivalent) than for crop cameras (which can use both EF and EF-S/crop sensor lenses). Lenses for full frame will generally tend to be bigger, heavier and more expensive. Sometimes much, much more expensive. For example, I use a 300/4 IS a lot with my crop cameras, shooting sports, action, wildlife, birds, etc. That's my longest handheld lens. If I want to enjoy the same "reach" with my full frame camera, that means getting out my 500/4 IS and a tripod to put it on (or at least a monopod). That's a $1200 lens vs a $10,000+ lens (if both were bought new today)... for the same angle of view with the different camera formats. That's approx. a 2.5 lb, 3.5 in. diameter by 8.5 in. long lens vs an approx. 8.5 lb, 5.75 in. diameter by 15.25 in long lens.
You also won't see some of the benefits of a full frame camera unless you were planning to make big prints... really big prints! Today's crop sensor cameras... and yours is still current... are really quite good. They have plenty of resolution for 13x19 prints... maybe even larger.
I use one, so can tell you for certain that the 5DII is not a great camera for moving subjects. It's close, but not as good tracking movement as your T2i/550D. If you want to shoot moving subjects with a full frame camera, the 6D is a better choice, though it's not as good as the 5DIII. In the crop cameras, all the Rebel/xxxD series except for the T3 are better, the 60D and 7D are much better following movement. I switch to my 7Ds for sports, action, wildlife, etc., both for the more responsive focus and for the "extra reach" (as described above).
The 5DII inherited the original 5D's AF system virtually unchanged. It got an upgraded processor that might have helped the AF perform better, except the 5DII also got 21MP, 14 bit resolution (vs 13MP, 12 bit in the 5D), so the bigger/faster processor is mostly occupied with handling much larger image files and doesn't help AF much. It was a shame and a disappointment, because Canon has already significantly upgraded the AF systems on the 40D and 50D, prior to the 5DII's release.
On any of the cameras mentioned and the one you have now, the quality of your images and focus performance might be more improved by getting different lenses or adding to what you already have, than by changing camera models.
My first instinct is to suggest that one year using your T2i/550D is really not enough to learn to use the camera well... That you should keep using it and, if anything, add a lens or two or upgrade some of your lenses, improve your image processing workflow, perhaps upgrade your computer monitor, software and calibrate your system. All these will likely give you much better improvement in your results, than jumping to a new camera model and starting the learning process all over again. An exception might be if you already have a fairly high quality lens kit and a lot of experience with this or other DSLRs and a high level of expertise using them and post-processing your images in a well managed workflow.
EDIT: I see you've added some info about the lenses you have... that's great! The 70-200 is plenty sharp and top quality with quite good AF performance. The 50/1.8 is a cheap, entry level lens with sketchy AF, but capable of making far better images than it's price might imply. The 18-55 is also an entry level lens, with simple micro motor focus drive.
Can't say for sure without looking at some actual image examples.... but be careful about evaluating images. What first appears to be missed focus might actually be camera shake from using too slow a shutter speed. An IS lens might help with that, but won't freeze subject movement as well. A monopod or tripod also can help, but also can't freeze subject movement.
I would suggest you consider better lenses, not a different camera. Get the 10-22mm, if wide angle interests you. Get a 17-55/2.8 IS. Eventually upgrade to the 70-200/4 IS or 70-200/2.8 IS, if the larger size, greater weight and higher price of the latter is okay with you. Lens upgrades will make far more difference in your images, than a camera upgrade will.