1) Read Genius, by James Gleick. It's a biography of Feynman which mostly concentrates on his life story (with just enough of his work thrown in to keep my inner physics geek happy) which happens to contain a couple of intriguing hints as to how he thought. Reading between the lines, I suspect he was (or had become) synaesthetic - he describes formulae as having colour, and combining them as entities rather than as series of symbols. He also mentions not understanding how his students comprehended equations if they could not see them this way, so he clearly understood that there was something different going on in his own head.
If that's true, it's a hint as to how he was such a formidable force, because somehow his sensory and proprioceptive apparatus had arranged to map directly to mathematical concepts such that intuitive manipulations which the brain is very, very good at doing anyway (because they are what's needed to drive the body and survive day-to-day) generated useful, valid, mathematical results. Or possibly, given descriptions of how he worked from other people, they generated many more options, whether valid or not, in a shorter time than anyone else, which meant he had a larger sample of possible ideas to pull the best one from.
Whether that translates to other geniuses or not, I don't know, but when people refer to "genius," they are referring to a qualitative difference in thought patterns. Where conventional thought might be verbal, symbolic manipulation, they are somehow using a higher level of abstraction. Finding a mapping between subject-specific concepts (like formulae, or chess positions) and something else your brain does "in the background" would seem to fit that.
2) No. I am not a genius. But I can do some things faster than other people.