uarghhh, your demands are very reasonable. Your buggy looks like it would be a straight forward conversion. It's hard to tell from the pics, but is that a metal or plastic spur gear? If it is plastic, you are good. All you'll need to do is fabricate a motor mount and use a mod1 pinion gear, assuming that your spur gear is mod1 (module 1) pitch. If the spur is metal, you will need one of two things. Either a hardened steel pinion gear, or a Kyosho plastic spur gear. The Kyosho plastic spur is the best option. It also fits many different diff cups. If it doesn't fit yours, you could try the diffs from Sportwerks, Ofna, and Hotbodies to name a few.
As for your electronics, it really depends on your budget. If you want the best, I'd say go for a Quark Monster Pro with a Neu 1515 1Y (for speed) or a 1515 2.5D for something a bit less crazy. If you are going with a 5S Lipo pack, I'd go for the 2.5D. If you are going with the 4S lipo pack, I'd probably go with the 1Y motor. I don't own the 1Y, but I do own motors of similar ratings and it would be fast without being stupid. If you want something less expensive, I would go with a Mamba Max speed controller and Feigao 8XL motor. Geared at about 14/46, it should give you what you are asking for.
For 10-15 minutes of runtime, just go with a 5000mAh battery pack.
As for the motor mount, if you have access to some basic power tools, like a drill, dremel, hacksaw, jig saw, or any combo of these, you could probably make a mount.
This was the first mount that I ever made. I used 3/16" thick 6061 aluminum. It is strong and easy to work. I now use 1/4" 7075, but it is a lot harder to work with. 3/16" thick 7075 would be a good compromise.
All this mount is, is an aluminum plate with holes cut into it and it is bolted to the center diff mount. It is very rigid and survived many crashes.