Two week notice

Probably...I was told that four weeks was customary notice, but they could not stop me from giving two weeks if that's what I chose to do. They can punish you in discretionary ways such as not paying out unused vacation or sick time, but I did not have either banked so it was not a concern. Nothing is stopping you from applying but more than likely unless you had a outstanding relationship with this manager they will not rehire you.

Those decisions - about how to categorize your termination & whether to put you on the "not eligible for rehire" list - may not be your manager's. In many organizations, HR does that. They make those designations based upon the established criteria. Unfortunately, some managers may not even know that this is done until after the fact.

CAN she? Probably. WILL she? It depends.

If your company's policy requires more than a 4-week notice of resignation in order to be considered separated "in good standing", then it's entirely likely that either your NM or HR will indicate your resignation and last shift dates in your file as a black mark against you.

It's also possible that you presented such extenuating circumstances (and if you were an exemplary employee) that your NM felt it reasonable to allow an exception; that might not be her call, might be HR's, but let's face it: they talk to each other. Not a great employee? I wouldn't count on getting favors.

Good luck.