Why are you guys subject to a credit check for a job? This makes no sense to me. Do they also inspect your house to ensure there's nothing untoward there? If not, they will soon by the sound of it.
You certainly don't have to allow them to do any sort of checks on you, this is after all private employment between two private parties (assuming non gub'ment job). Everything is negotiable.
I personally have never taken a drug test as part of a hiring process. I've been with my current employer about a year. I've only once had a credit check and that was probably five years ago with my previous employer.