OK, here's the problem:
This is not the usual "black screen" problem you can find solutions on google for.
The game doesn't crash and it loads just fine, only when in game every time there are certain light effects (like lightining or some explosions) the screen goes completely black.
I had the same issues with the first game but there you could just force DirectX 9 but here when I do that the image on the screen looks "shaky" and fuzzy even though it's set to 1080p so it's not an option.
Also yes, I've already set the game to use the 64 bit executable.

I have Windows 7 64 bit, i5-3470, 8gb of ram and a Radeon RX 580 8gb.

I think I have solved the problem: it seems the game didn't play nice with the shader cache option in the AMD Crimson profile being activated.
The parts that realiably made the image disappear from the screen now behave as they should, both in Warhead and in Dx10 Crysis.

I went on and finished Warhead and I noticed that I still got the occasional black frame usually when I got caught up in an explosion but nothing as game-breaking as before deactivating the shader cache.
It is a weird bug though, I wonder what exactly is causing it since I don't find anyone else on the internet with the same issue.