Openings for 'experienced' beginners developing their game further

We could debate on what superGMs are doing. However, I think I preempted that by saying that GMs don't do anything weird in the beginning phase of the opening. If you look at the Italian, with all its fascinating tempo play, you'll still see that the first 5 moves are almost invariably made up of moves that are following the opening principles. Yes, after move x they might start doing superGM stuff with their openings, but they start out the same way as we do. Anyway, for us mortals that's not so problematic, because we can still play for all results when we do play normal moves instead of computer generated ones.

I disagree. Stronger players don't usually win by going for traps. That is because going for a trap usually would be risky. Stronger players will have a big edge in the middlegame anyway, so usually they would try and beat you there. That being said, it's not a bad idea to try and learn the opening traps in your opening. In fact, if you would play 1. e4, e5, I would advise that you did learn about some traps (or make sure you learn the first time you lose against one of them).

I am fine with some basic opneing principles like moving pawns to the center or getting your knight and bishop out but that is obvious to anyone who has not even played chess before. The only thing one can do in the opening which makes any sense is move the pawns and get bishops and knights out or atleast the knight.

However focusing on specific opening theory is useless. Any time spent on studying an opening or building an opening reportoire is time wasted for amateur players. This is true even for strong club players(close to 2000 rating), but it's even more true for beginners. They should just try to calculate and visualize better and most of their problems will be solved

There is no point in memorizing 15 moves of the Winawer French if your opponent deviates on move 3, or - given that you find someone who is willing to enter a theoretical battle - blunder a knight on move 16. 

And since everyone seems to talk about Nelson, I tried him too. It went like this:

He plays rubbish. Absolutely no opening theory required (I was out of book after move 2) to beat him, just a few logical moves.

As a beginner, I just feel like I want to learn a minimum in opening because it's kind of reassuring to know a few moves (may be just the 3/4 firsts moves of an opening) that is "theorically a good way to start a game" (I'm speaking about the openings that we (beginners) can easily understand as respecting the fundamentals principles of openings, not the too complicated ones). Even if we often have to change plans anyway...

About the importance of focusing on middlegame, I regularly see this advice. But what are the principal way/tools to train, learn and progress on it ?

Is there famous books, exercices or anything else ?

PS : I know nearly nothing about openings and I can beat the Nelson bot, only because I learned and understand the weaknesses of his queen by playing him.

Yes, there are many good books on strategy and middle game planing.

Watching / Playing through master games is helpful too.

Yes, there are some very good books on the middlegame! Tons of chess videos also focus on specific middlegame strategies. I always liked Silman's book "How to reassess your chess" and have used it when teaching my pupils. That being said, some people have also pointed towards other books they thought were easier for lower rated players. Maybe it's worth it to search through the 'beginners' forums' or start a thread about this if you want more good answers.