Use Game Trailer Videos?

If you have commercial intentions you can download the YouTube Game trailer from e.g. CallOfDuty, csgo, Minecraft ect, cut it cool and add music to it and use it as a background video on your own website - or it is generally prohibited.

At Hytale, for example, you can use it for commenting.

Does anyone know more?

You have to observe the copyright.

With the permission of the other rights holders, you are not allowed to do so, otherwise.