I don't think it'd be rude per se, but I think you'd be hard pressed to have the professor agree just because you have other exams. It's not an emergency. Others have multiple exams going on too. Such is life.
Most of my syllabi explicitly say the multiple tests are inexcusable for test date changes. Check yours before making such requests. I think you will likely get a no, because like the above poster mentioned, everyone has this issue.
Yeah good choice. Our school had a policy where if you had 3+ exams in ONE day then you have the privilege of speaking to your profs and bumping an exam to an earlier date, assuming prior notice of course. If it's 3 in a week then that's considered pretty normal midterm schedule =D
What do you think happens during finals? During med school? USMLE? MCAT? "Oh AAMC let me take different test sections on different days, because it's hard and stuff."