Kodiak

This is also freerunning, there aren't any gymnast moves in parkour.

Nahh, you're missing the point. The guys doing parkour do it for entirely different reasons. Ask yourself... What is the difference between a martial artist and a dancer? On the surface - nothing, but there is something "huge" that makes the difference... do you know what that is? Answers on a postcard to... Sojobo No, JusticeZero has it pretty much exactly right. I suspect you're thinking of freerunning.

It's not about proving you're tough, its about having the same attitude of protecting your hands the old fashioned way at the bag that you'll need in a self defense situation.

See if you can't find somebody to run you through things and tell you where you're weakest. Your instructor may do this for you too in the coming weeks. In my experience if I don't specifically target weak points in my skills I train the things I'm already good (because well, I practice those things all the time). Don't stress too much, most place don't expect to see serious skill in low belts, they want to see if you can actually do the techniques so they can focus on new things, coordination, timing, intensity etc takes a lot of time, which is why the kyu/dan system exists in the first place.

Somebody has to tell you when you make a mistake, or you'll keep repeating it over and over. And unless you have a really good ability to copy moves, I doubt that enough detail will be in the video to do it right. Missing details can include safety things too.

Martial arts is hard enough without throwing in language and cultural barriers. There's probably people there worth the trip to learn from, but I think that would be true of most places I could visit (heck, the Grandmaster of the organization I'm currently studying under lives in Evanston, Wyoming of all places).

Clubs.

I like Evergel wraps. no restriction in how I move my hand, thin padding, and build in velcro straps. They even have padding along open handed strike points. TKD gloves look similar, but I've not tried them. I do occasionally do gloveless work without serious injury (I recommend against this if you've ever done work with the assumption of wearing gloves or have less than a year or so of technique practice), but I lose skin doing it, so its really limited to conditioning work, I can't do serious work on speed and power without the gloves.

Being able to take more advanced black belt only courses. Oh and ego and pride and all that stuff.

I'd worry less about the style and more about the instructor. Do a trial period at a few places (if you can) or at least sit in on a class or two. Figure out what kind of instruction works for you, and if you can, find an instructor who carries the style, rather than somebody who lets the style carry them.

As somebody whose recently restarted in a new style, there's a lot you have to get out of your head in terms of how you used to do things. While I do try to use all that old stuff in my practice at home, I'm not going to learn anything new if I don't put it out of my head at the studio.

I'm looking for an appropriate martial arts shoe (something similar to Feiyues ideally), however I have extra wide (EEE) feet. Does anybody know a company that makes extra wide martial arts shoes?