Is the ambiguity of a regular tree grammar decidable?
Start by considering regular string grammars. We can determine whether one such grammar $G$ is ambiguous by constructing the intersection of the grammar with itself, with a direct product construction. The nonterminals are pairs $(A,B)$ of nonterminals from the original string grammar $G$. The new grammar of course also derives the original language $L(G)$, but the new grammar has a nonterminal $(A,B)$ with $A\neq B$ in a succesful derivation, iff $G$ is ambiguous. It is decidable whether any given nonterminal occurs in a succesful derivation, so ambiguity is decidable for regular grammars.

The same is true, mutatis mutandis, for regular tree grammars.