Decidability of empty intersection of two languages accepted by Turing machines
It is undecidable whether a Turing machine accepts any input at all (reduction from the halting problem). So, take a machine $M_1$ that accepts all inputs. $L(M_1)\cap L(M_2) = L(M_2)$ so non-emptiness is undecidable.

The intersection of two RE sets is RE. This is a standard fact: simulate the accepting machines in parallel and accept iff they both accept.