Okay, reboot isn't quite gone but it's not frequent anymore... I'm starting to think the problem may be with the sd card. My brother has the Galaxy S2 and never had a reboot, his phone doesn't have an sd card as he never got round to buy one. Going to test without an sd card first and than with a different brand and volume to see if it's brand specific or that the sd interface is at fault here.
One thing I know for sure is that it's NOT the battery! I put in a brand new original one in the phone and after a few weeks the first reboot happened.
To sum it up: my S3 mini's battery does not cause the reboots and all seem to point to the sd memory card and/or it's interface. This as far as I could figure out.
I'll keep you posted as I see new things happen, or not.