First of all, I am not a lawyer and in this site questions about these issues have to be always taken as ideas, thoughts or experiences, never as technical advice.
Second, I see you are not from US and I have to tell you that what I am about to say is fairly based on the US laws.
So, that said, you don't need, but you can and you likely should copyright the specific parts of your game. Let's see.
we are working on the name of the game (just use temp one for now)
Game titles are usually not copyrightable, but they can be protected under trademarks. Those are different things. But yes, you should trademark your game title when possible.
Do I need to protect everything separately?
Kind of. Code is copyrighted similarly to literary works, although there is quite a debate on to which extent code can/should be patented. While art and sound are usually copyrighted as audiovisual.
Can I use my already made domain and subdomain for the game? Or do I need to register unique one domaine
If you are talking about your website, that's your choice. As far as my experience with that goes, your game doesn't even need to have a website, and it certainly doesn't need to have the same name as the game. You can have a website like www.mycompany.com/mygame. No problem with that. It's your choice, not a legal requirement.
Now, about your more general question about whether you should or not pursue legally protecting your game and its pieces, I think at some point you should. There is quite an interesting article on the NewMediaRights website, on copyright, trademarks and intelectual property for video-games: http://newmediarights.org/guide/legal/Video_Games_law_Copyright_Trademark_Intellectual_Property
As they say, "While you don’t need to have the work (ie your video game) registered to covered by copyright law, there are advantages to registration (see our guide to copyright law)".
So, what are these advantages? The most obvious are: first, it is much more effective in scaring people off trying to use your stuff without authorization. Second, it makes eventual legal battles easier to win. Third, it saves you the eventual legal battle part of proving you did create what you are saying you created. That saves time and money. Fourth, it is cheap, as already said in other answer. Among other things. The NewMediaRights has a very good guide for Copyright as well: http://www.newmediarights.org/frequently_asked_questions_about_copyright_law_book
You will be particularly interested in the section "IV. Now that I have the copyright to something, what does this allow me to do?".
For a basic introduction to the questions related to copyright, trademark and etc, you can also try the Gamasutra article: http://www.gamasutra.com/view/feature/3546/hey_thats_my_game_intellectual_.php?print=1
But of course, I don't want to mean you have to be over-concerned at the very beginning. I think that this is the kind of problem I start thinking of more when the project has developed quite a bit. After all, as I just said above, at least in the US and countries with which the US has treaties on these things, just by making something you already have a set of rights about it, even it you didn't copyright.
Still, if you made graphical or sound assets that, in case your project is eventually abandoned, you would still try profit on somehow, that's a stronger incentive for protecting them from the beginning. You would be surprised how often people take other's assets without permission even for professional-ish uses. Not to mention how many assets are put online as if they were copyright-free, when they are not.