If a theorem holds for all truth functions of all n-valued logics, in all n-valued logics with quantifiers, n>=2, will it also hold in classical predicate logic where the domain has at least two elements? Does the converse hold?
Yes, bi-valued Boolean logic is a special case of many-valued logic.
No, e.g. in Lukasevichz logic $p \lor \lnot p$ does not hold but it is correct in bi-valued Boolean logic.
See Petr Hajek's book "Metamathematics of Fuzzy Logic" for more information.