EDIT: The problem is partially solved. I tried unplugging the SATA DVD drive and now the system boots just fine. Does this mean that that the DVD drive ist busted? Is it normal for a faulty DVD drive to have this effect?
One related thing might be that my DVD drive has disappeared. The ''open drive'' button doesn't work and it's nowhere to be seen in the device manager either.
EDIT: I don't know when this happened. I barely use the drive so it might have been months ago or a few days ago.
Another older problem that probably isn't related, but might be: Windows 7 has installed the same two updates again and again for the last few months and never recognizes that they are installed. They are KB2742595 and KB2789642.
120GB A-Data S511 2.5" (6.4cm) SATA 6Gb/ MLC asynchron (AS511S3-120GM-C)
8GB (2x 4096MB) G.Skill NT Series DDR3-1333 DIMM CL9-9-9-24 Dual Kit
1024MB XFX Radeon HD 6870 Dual Fan Aktiv PCIe 2.1 x16 2xDVI/1xHDMI/2xminiDP (Retail)
Disconnect the DVD drive, check that your boot drive is first in the boot order.
The DVD drive was behind the SSD in the boot order. I disconnected the DVD drive and that worked. Removing the DVD drive from the boot order also seemed to do the trick. That made the DVD drive work within the OS. I guess the problem has been solved, even though the root of it is yet to be found. My GPU randomly peaking while the computer is idle is however pretty annoying. I guess I should start a new thread about that? (After I try a few things first of course.)