1. Do you mean to say that when the faucet handle is moved to the off position that the water continues to flow or drip? If so then just tell the plumber, "My water continues to run after I turn the handles off." Or do you mean that the handle itself will not move to the off position (and the water either runs or not)? In either case, if these faucets have been working fine before, it is not likely that anything is stripped.

2. How can the handle not move but something else does? Are you saying it just wiggles but will not rotate? Has someone already tried to fix this? I just can't imagine how normal use would break a faucet the way you described. How old are the fixtures? From what I'm imagining your problem to be, I would tell the plumber, "I need a new faucet." You will have to pick one out or he will bring a very inexpensive one that is easy for him to install.

1. either a rebuild kit or a couple of 5 cent washers will fix this. shut off water to teh house, ope a faucet or two(one on low floor one higher) to relieve pressure/drain the lines down,remove the handles, pull the stems and replace the washer or replace with new stems is teh handles are starting to strip. once everything is reinstalled, turn teh water back on to the house and check for leaks.

2. replace the faucet. if it is spinning like that the housing is broke.

no need for a plumber on this usually, that would cost more than the parts to DIY it, and this being a simple fix you should try.