Is checking if regular languages are equivalent decidable?
It is decidable, all you have to do is use the closure properties of finite automata + emptiness checking; let me know if you want me details.
I’m not sure exactly how you’d check the equivalence of two regular expressions, other than by converting to automata, so unless this is a practical issue you’re trying to resolve and you receive input in terms of regular expressions, I’d think about it in terms of automata.
As for creating the Turing machine that actually does this, it’s probably not strictly necessary to write out the formal description, as long as you can argue why it’s possible to do so.