What kind of dental work is needed for a filling falling out and tooth above and around the filling breaking off?

Possibly: A root canal but more than likely a crown as well.

It depends: It depends on how large the original filling was, how extensive is the breakage of the tooth around the filling, and whether there's decay, pain. It could be as simple as just replacing the lost filling with a new, larger filling, or it could go all the way to a crown and possibly a root canal as well. Worst case scenario, if the breakage is large enough, the tooth may need to be extracted.