Is there something like a health requirement in serious airlines? Or is it just a myth? How are potential pilots screened when it comes down to physical requirements?
Here are the mouth and throat requirements for a pilot physical.
The main thing is that it shouldn't affect your ability to fly (i.e. having pain) and ability to communicate over the radio.
Commercial pilots who don't fly regular schedules need a Class 2 medical certificate. Regular schedule commercial pilots need a Class 1.
The major requirement for dental fillings as far as aviation (or diving) is concerned is there not be a trapped air pocket under the filling (this could cause problems for divers or pilots as the pressure changes may cause discomfort. (It's called barodontalgia - pressure-induced tooth pain.)
Air pockets are something your dentist will try to avoid anyway because they promote further decay & will eventually require more work on the tooth, so if your dentist is competent you shouldn't have any problems.
If you do notice discomfort while flying mention it to your dentist - an X-Ray can usually identify the problem in a few minutes, and your dentist can then take care of it (replace the filling, or if it's a new cavity treat it appropriately).
Dental fillings, false teeth, implants are all permitted so long as the mouth is healthy and there are no abcesses.
Commercial pilots need a first class medical certificate. I don't have that reference here with me now, but perhaps someone can post either a link to the requirements for a first class medical, or list the requirements here.