You have to read the entire procedure in the code to see that the slate from the Executive of the state is the default in case of a tie. Multiple slates may be sent by a state, e.g., the complicit Governor's slate and any slates from the Legislature, depending on their state election laws.
Correct, that is, if the governor sends a separate slate (which has indeed happened a couple of times) THEN it is the job of the Congress and Senate in closed session to determine which slate to accept.
Pence has no role, except to say, “There is an objection. The houses will now retire to address the objection.”