Road bikes traditionally have a better BB, mountain bikes, hybrids, cyclocross bikes tend to have a higher BB for clearance. Bikes with a lower BB handle more nimble, due to the lower center of gravity. Unless you're a total klutz, to me it comes down to where you are riding, not how well you ride. But, because bikes with lower BB are more nimble they are quicker to turn or lean, so that could be an issue.

It's hard to advise on components, better components are lighter, and usually perform better. Without knowing your true ability, it's really about budget. You can always upgrade your components if they are not sufficient.

No matter what level of rider you are, you are cheating yourself if you skimp on rims though.