“3. What authority does Congress have to compel state legislatures to do anything?”
Well, Congress can just say “do what you like, but without clarification, we can’t count these electoral votes”. Pretty sure most states would find that a compelling reason to do something.
Read the U.S. Constitution. Congress has no authority to do such a thing.
You give Congress the power to do this, and fascists like Nancy Pelosi and Chuck Schumer will be refusing to accept electoral votes from Republican states for any damn reason they wish.
And if a state legislature convenes in a manner that violates its own STATE constitution, then any action it takes isn't legitimate anyway.