I prefer regular shoes over sneakers. My sneakers are b-ball shoes so they are kind of heavy and thick and don't allow too much feeling to my feet.
Putting the car in neutral and using the brakes to stop is for noobs. I downshift all the time when coming to a stop. Do you heel-and-toe downshift? That will allow for shorter stopping distances. I used to practice it, but I'm very shitty at it so I just stopped. Now if I wanna use my brakes and downshift somewhat at the same time, I push the clutch in, downshift and blip the throttle, and as soon as I blipthe throttle I immediately step on the brake pedal as I am letting the clutch out in the lower gear. This is as opposed to braking AFTER I've let the clutch out....this way seems to save me a lot of time, however I have to make sure that I blip the throttle hard enough because if I mess up, my foot is already on the brake pedal so I can't really "correct" the mistake.
One technique that I've been told by a really great driver, if you need maximum acceleration, when you downshift you can blip the throttle a lot harder than you're supposed to and let the clutch out real fast while applying gas and since you over-revved the car will do like a little "jump" forward for a nice burst of acceleration. I don't use this technique on my car as it obviously burns the clutch just a little bit, but I'd imagine that on a racetrack or something it would be more useful. The technique is not to be used all the time, just like on a downshift on a straight-away....if you tried doing a downshift like this at the entrance of a turn it could totally mess you up.