Report of my encounter with an Apple Genius. Good news is, it's not a hardware problem but software. The installation of an app messed up the permissions too badly/agressivly/deeply. Something went wrong somewhere. Unfortunatly there was no way to tell which application caused that.
One command line I saw him tried was 'sudo rm -rf ~/.Trash', but didn't solved the problem. They have multiple tools at the Apple Store unavailable to us and none of them worked.
First thing that tipped him off was when rebooting in safe mode (holding shift key while the computer starts), there was no problem. Safe mode, if I understood him correctly, starts the OS with just the essentials to be running. There was no problem with the essentials, so it had to be one of the program I installed. Second thing he did to confirm is creating a test account to see if the problems were still there. They were not.
Two things were possible to solve it, nuked the HDD and start over, or create a new admin account, transfer everything over there and delete the first. I chose the second option. Now everything works fine.
If I had to guess, it's probably Git (dev tool for version control) that messed up something. I am now extra carefull before installing it. Soon I'll even install the OS on an external drive and test everything on it before intalling something.