What's wrong with my card?

Either it is a heat problem or the card is gone bad. When you removed the heatsink the first time did you use new thermal paste on re assembly?

The first time a few months ago, no I did not. I put it back as it was and worked immediately. I just removed the dust which was enough. But now, after I tried it a couple times with the old paste and didn't work, I removed the old paste completely very carefully and applied new paste.

The weird thing, is that the problem appeared when I tried to install and launch a new game. Since then, the problem disappeared once and I was able to use the card with my old games, but I tried again to launch that game and since the second time, it is not working. I tried several time installing and uninstalling the drivers, I even formatted my drive and tried with a new installation but nothing.

I would say that the card is dead (or dying). Always when removing coolers new paste is required otherwise you can not ensure that there is enough heat transfer from the GPU (same applies to CPU´s) to the cooler.

Probably it is dying.. But the first time when I removed the cooler and put it back without applying new paste, it worked for another 4-5 months without problems until now..
Why without drivers the card works with 640x480 resolution and 4 colors? Isn't that weird? Shouldn't be not working at all?

With no drivers it is limited to 2D which is really low load, defects only show when under load in 3D.