Most ticks bite, fill up on blood, then drop off - since you wouldn't be carrying anything for one tick to mate with, that would also be why it wouldn't stick around. They lay their eggs on the ground, those hatch and go through larvae and nymph stages. It's usually the deer tick nymphs that spread Lyme disease as well.
Some people get the red bull eye rash and some don't.
I suggest you go to the Lyme Disease section as it will have more information on exact tests - which one is more affective, false negatives, etc.
As for how common - it varies by area. I also suggest you go to the CDC's homepage and look at their Lyme information as buried somewhere in there are statistics of Lyme Disease reported by state - and by year, month, sex, and age.