will my school excuse this?

It's only a few days don't worry. Maybe try looking into bus or train routes?

Probably not. It was your mother's responsibility to contact the school and to arrange for transportation PRIOR to the surgery. If the surgery was an emergency procedure, you or your guardian needed to contact the school.

Is the U.S., students at public schools either live close enough to walk/ride a bike, or a bus is provided. Kids who attend private schools can often take public transportation or get rides from classmates. Missing one day would certainly be understandable, but missing five is not.

Before the surgery your mother should of had a meeting with your school to let them know and to discuss option to get you to school.

Doing this well after the fact will cause problems.

I am surprised the school didn’t call your home by the second day

if you told them you would be missing that much, usally they call to find out why you are not in school, or just have your mom call an let them know why

In the schools that I'm familiar with you're expected to walk if you don't have a ride and you live too close for bus service.

Maybe it's different in your school.