This is not accurate. In Germany, you usually only have to get a doctor's note at 2 or 3 days, if youre only sick for a day or two you don't need one.

And there's an unlimited number of sick days. As long as you have a doctor's note, you still get paid, up to some ridiculous limit at which you might have to get government support instead.

It's up to the employer, they can ask for a doctor's note from day 1. Many employers have more lenient rules, though.

I think at some limit the health insurance pays back the employer, right?