First Question: Yes.
Second Question: To make the automaton of part (a) totally defined, you just need to add a sink state as you did. However, part (c) also asks for a symbolic encoding of that automaton, and for the three states that you have, you need two state variables which in turn introduces a fourth state that must also have outgoing transitions for all inputs. So, you cannot solve the entire part (c) without the fourth state.
Third Question: A precise algorithm to check equivalence of the automata is difficult. You would have to determinize them and then you would have to check the equivalence of their acceptance conditions on their product automaton. While that can be done, it is certainly not recommended for the exam problems. Here, you better argue intuitively by reasoning about the words accepted from the states.