I don't think you should have her stay with you. I could let her know you're all very run down and life is chaos so she'll need to stay somewhere. And probably staying closer to the city is better due to her early appointment. If she does stay with you then no way do you have to go to her appointment with her too. She's a grown woman moving overseas, she can manage it. Yes, the US embassy generally only makes early morning appointments. They put a bunch of people on one time slot and then you hang around while you get it sorted. They end up with people there for hours so they don't make late appointments.

Personally I would go into the city to see her after her appointment for lunch or something like that. Or have her come and stay a night with you after her appointment.