Rubar, welcome to the wonderful world of image capture and sharing - it is a lot of fun. And THIS site/community is THE best site on the web.

To your questions.
image quality with extender at full focal lengths? Plenty fine - down right excellent
auto-focus speed that by spec will be slower? Is it really noticeably slower? not enough to be all that worried about and you will have to study to find a meaningful difference in all but lowest of light that no camera is all that predictable in - you will find the center point focus works most the time anyway.

Neuro makes the point on the focusing is always done wide open, so don't worry about the stopping down event.

Wildlife - depends - some cheetah chasing a springbok while you are shooting from a bouncing land rover - focus won't be the problem - finding the target will be.
Anything that is predictable or moving at a modest pace this combo will be fine. Birds in flight (BIF) are simply really tough to even find on a 640 mm equivalent lens, let alone track, plan for the wing beat vs. shutter lag - just how many shots like this are you going to try and take? Even someone using the 1dx needs a lot of practice to get it right.

If you think of all your pictures you are likely to WANT to take as an envelop - like a flight envelop of a plane - how dark/bright, moving targets, sequencing between shots (Frames per second), non-moving targets, macro (extension tubes are a blast w/ the 70-200), etc, then imagine how far out on each of these axis your camera/lens combo can take you. You won't be limited by the 70-200 lens & 2x!. When you look at the likely percentage of shots the next camera model "up" gets you, you will find you have a darn powerful set up. The 70-200 is a world class set of glass. Canon makes nothing better in a long zoom.

While I am on it - PLEASE learn how to hold the rig correctly - left hand takes ALL the weight, right hand drives the buttons - they put the tripod collar on the lens for a reason. Elbow against your side - you become a virtual bi-pod. If you are trying to be really still relax your shoulders and tighten your butt & gut - try standing on one leg with alternating tightness, it is crystal clear where you get best balance. Yep the 2.8 is heavy - so what - get a decent tripod/monopod if you are going to be standing for such a long time.

2nd to last - Practice, practice, practice - if you wear out the shutter in the 1st year Canon will replace it under warranty.... I doubt that you can though. Any given 100 exposures is effectively free. While the 1dx can fire off 14 FPS, you will be a better shooter if you plan, lead, and fire 1 shot at a time. Go to a local soccer match and try manual focusing - you will learn which focus modes are really for what environment more quickly. Did I say practice yet?

Lastly, if you haven't downloaded Magic Lantern and put it on your camera do that next - it expands the functionality of the t3i to an amazing level. While you can do this for free - toss the ML group $10 bucks - they are planning for some cool upgrades and man they deserve ever penny they get.

While I have been infected with the pixel peeping version of the GAS "Gear Acquisition Syndrome" you won't be disappointed.

Happy pix

Did I say have fun yet? I didn't for crying out loud go have some fun with this stuff.