Problem after Installation with my SSD and the whole GiB

You're contradicting yourself. Either you want Zorin to use the whole disk or you don't. Which is it?

Nothing is being damaged by Zorin, nothing.

Well there's part of your problem right there by your own words that has nothing whatsoever to do with Zorin. Your hard drive already has bad sectors.

I'm not sure why you felt the need to "clean" the drive before installing Zorin. Perhaps whatever you used to clean the drive also broke the drive.

When installing, you should choose something else and tell the installer which partitions to make for install and how much space you want it to have. But even if you do get it installed this still doesn't address the fact you have bad sectors on a drive. How old is this drive?

My suggestion to you is that you get yourself a new hard drive.