Tree automata with constraints on infinite trees

Fix a finite alphabet $A$. Denote by $T$ the set of $A$-labellings of the complete infinite binary tree, i.e. maps $\{0,1\}^*\rightarrow A$. Let $R \subseteq T \times T$ be an (equivalence) relation of such trees. A (binary) tree automaton with $R$-constraints consists of a finite state set $Q$, an initial state $q_0 \in Q$, and a transition relation $\Delta\subseteq (Q \times A\times \{R,\neg R\}) \times (Q \times Q)$. A run of a tree automaton on a tree labelling $\lambda:\{0,1\}^*\rightarrow A$ is a choice of states $q:\{0,1\}^*\rightarrow Q$ and transitions $t:\{0,1\}^*\rightarrow \Delta$ for each vertex such that $q(\epsilon)=q_0$ and $t(w)=(q(w),\lambda(w),X,q(w0),q(w1))$, where $X=R$ if the left and right subtree of $w$ are in $R$-relation and $X=\neg R$ otherwise. A tree automaton accepts a labelling if there is a run for it. This can be extended to different more intricate acceptance conditions (Büchi condition along paths, parity conditions, \ldots{}). The set of labellings accepted by an automaton is called its language. The following questions might depend to some extent on the nature of the relation $R$ and should be understood as "for your favourite/interesting $R$". Question: Is emptiness/universality decidable for these automata? Question: What are the closure properties for these language classes?