>Should I buy/register the domain name for my game website? If so how do I do it or where do I go?
You could, but a wordpress is enough. Make sure to put images and videos of the game. And make them as good as possible.
>I have some experience in web design, but any recommendations about templates or something that is better for games websites?
Find one with easy to read information. Is not about the template, it is about the game.
>What about normie places like facebook or twitter? I don't even step in those places but should I got on it, and what kind of approach should I have?
I don't think everygame need a page on facebook and twitter, but it helps a little bit depending on the game.
>I'm not even selling my game for money since it's my first project, but I guess I should at least put a pay what you want button or something like that for all the effort it took me, but how do you even do that?
Are you talking how to implement that on your site? Or game design? Donate buttons are pretty easy to make via PayPal.
>Any specific recommendation about how to sell your game or get close to your target?
Steam, forums (Game development tool you use) and Steam forums(your game forum) are a good start.
>What about Steam publishing, how hard is to get into greenlight?
Nowadays is pretty easy. There are communities that will help you if you help them to greenlight them. Also, a lot of good people look into greenlight. And various youtubers make series about greenlight games. Even if your game is bad, people will hit the yes button.