Some people say that Mojang said they're cool with it, but in their terms agreement, it sounds like they're not at all cool with it. I make seed videos and want to know this.
Above all, you want to make sure that you own the right to use every single aspect of your video - Including thumbnails - Don't go on Google and randomly download a thumbnail template that you like. Make sure it is royalty free. The same goes for music. And as for minecraft, it would not be where it is right now without people doing their lets plays and awesome creations on YouTube.
But in simple, yes, you can monetize Minecraft.
As far as other games, that's a grey area, and depends on the content type
eg: if you write a book with a pen, would you be the writer of the book or the pen? Consider Minecraft a tool for making 'your' videos