How to get rid of black triangles and artefacts? for the past few days, I've been struggling against artifacts problems with a mesh.

I used the boolean modifier on this following cube : 



After applying the "Difference Operation", here is the result : 



At first, It looks great, but if I zoom in, you will notice this problem :



I have tried to use the Edge Split modifier, It did nothing.

I also tried to remove doubles, and this happened : 



Not only It accentuated the problem, but It also created some holes (as you can see at the bottom left of the picture).
My guess is that some faces are overlapping. This problem happens with all the rounded edges.
Do you guys have any tip on how to solve this ?
Solved !
Credit to @Hikariztw and @Duarte Farrajota Ramos for pointing out the answer ! 

Sadly, I was only tweaking the clip start and not the clip end. 

By tweaking BOTH of these options, here is the result :



z-fighting no longer showing up.

Thanks a lot for the help !