Okay so I did as you said, checked every connector and reseating the heatsink/fan of the CPU. I also moved the case fan from the bottom front (where it was when I got the case) to the top, at the back. I turned it around so it will suck the hot air out of the case.
But now I've got a new problem: when I connected my PC to my monitor via DVI, I didn't get a signal... Sometimes the monitor would light up a little but then it went dark again. I tested it on my graphic card and onboard, both not working. I also tested a TV on HDMI, again onboard and on the graphics card, but nevertheless it didn't work.
Then I checked inside the case during startup, and all the fans and leds were working. I also heard a beep, but I'm not sure if that's the particular startup beep or something else, because when I restarted the PC without the onboard speaker connected, I still heard a beep... Also, when I open de DVD drive, it closes right away, and the case fan seems to not spin on the right speed and sometimes even really slow.
I'm pretty sure It hasn't anything to do with my PSU, because I replaced that recently. I tested the RAM and HDD/SSD without issues so I don't think that's the problem either.
What I also should mention is that I once did a failed BIOS update from destkop with MSI Live Update, but my BIOS did boot again so i had the chance to do a update from the BIOS itself. Maybe it could be that it still did some damage to my motherboard?