Did my graphics card (GPU) burn out?
With a bit of reasoning and the help of some users in the comment section of my question I quickly came to the conclusion that my ghraphics card just died of old age. Installed in March 2011, it lasted around 5 years, and that's no surprise since that I used to use it quite a lot to play computer games etc. The red lines and random colored pixels make me think of an hardware failure in the VRAM of the card, which is a definitely unsolvable issue. To test it even further I tried mounting my roomate's graphics card on my motherboard and it ran smoothly without even configuring anything.
I bought a new card, an Asus NVIDIA GTX 960 Strix OC 4GB, and I'll have it delivered in a few days.