What am I supposed to do when management throws a filthy piece of shitty code at me to magically figure out bugs in & render operational overnight?

Have some coffees and do what you can. If u are successful, they will take notice. If you don't they won't . Its ok to fail. You can't do everything on deadline just because someone wants it done. He will allocate more developers if he really wants it done in time

In my view just say no kids

Time for a throw-away-and-rewrite party!

Sort it out. If you don't someone else would probably be happy to.