I am by no means saying "don't do it", because it's an awesome idea and sometimes things just have to be done for the sake of doing them and experimentation. I was just pointing out the thoughts that came to my mind while thinking about the idea.
You'd think there would have to be some way to shield the electronic ignition components from the magnetism, even if that meant just putting them in a different position than originally antisipated. As for cell phones and credit cards, they're both big conveniences, but guys rode without both of those things for a long time, so who really needs them, lol. Or maybe carry your phone and wallet in your jacket(if you're wearing one that is) instead of in your pants.
As for heat, i'm not sure how much the engine heat would really effect something mounted under the seat. I'll admit that it's something i've never thought about, but it would seem to me that if the whole engine is air cooled, and you're opening the frame up even more to space(air), that it, and the magents, would have a decent chance of being cooled. Now if magents produce heat from being cycled(closer, further, closer, etc), then that could be a whole different scenario since they would be doing this quite often.
As for road debris, if you're speaking for an electo magnet, I can't really say anything since I have no experience with such things, but as for solid state, I don't think it would be that big of a deal. I've played with some small, but fairly powerful solid state magents, and i've never really had problems with them picking up random particles. They are quite powerful when repelling each other, but it's not like they're going to pick up a nail a foot away(not in the strengths I was playing with anyway). Obviously the less the mass of the object the further away it can be and still be attracted, but I would think at seat heigth, let's say you're going real low at 16", that's still a hell of a long way for them to attract random things. Plus if you were to go with a more "sealed" system like with the guide pipes, then you could just wipe any debris off.
As for size/mounting I just remembered an episode of Mythbusters where Jamie used large solid state magnets to pull himself up a metal wall. He used 2 magnets to do such a thing(but he would hang from one as he moved the other), so I would think that that would be the maximum size of magnets needed. Even if Jamie isn't that heavy, my thought process is this: 1 magnet has an attraction force of 150lbs(I would think Jamie weighs more, but the point is the same), then it should also have a repelling force of 150lbs. You need 2 magnets to make a floating seat, so if both are the same size, you now have a repelling force of 300lbs. This is my thought anyway, but I am not really into magnets or exactly how they work.
Hopefully that gives you some food for thought. If I were you, I might still try this. Like I said, I am by no means trying to discourage you, i'm just trying to present you with enough information/ideas to make an educated design/decision. I would think that the next step might be looking in a magnet supply catalog(do those exist?) to see what options are available because to me personally even with some of the disadvantages it looks like solid states are coming out ahead in this scenario.