15 sick days is very generous policy especially for a part time position. It’s not strict. Doctor note though is a bit much especially if they don’t provide health care. How would they expect you to pay for appointments? Could you ask for details re doctors note?

I can’t say I was ever treated like garbage but I had some bad bosses. Honestly if you need a job in a hurry, you will have to put up with some nonsense if don’t have too many options. I’d give a try.