Choice of Opening

Well, I do 'know' the principles. Can't assure I understand them always, because sometimes I mess up even though I try to follow them, specially when my opponent responses are not usual. I mean, like d4, b3, g3 as white, e6 or b6 or even Nf6 as black, I feel uncomfortable against these.

I invite you to check out my thread where I suggest a good option if you're looking for something easy to understand and that doesn't require much study, allowing you to manage without doing much.

https://www.chess.com/forum/view/chess-openings/easy-opening-setup-fer-very-beginner-players-and-lazy-players

You should not be studying openings at 700. Endgames, Tactics, and Strategy. Apply BASIC opening concepts - control the center, don't move the same piece twice, get castled, do not move the Queen early.

After you reach about 1600, look at the games you have played. Did one opening setup make more sense to you than others?

In 1995, when I knew the rules of chess for 12 years but never read a chess book, I read Winning Chess Tactics, Winning Chess Strategies, and How to Win in the Chess Endings.

During that time, I experimented with opening ideas. Always tried to fight the center, meaning ...e5 or ...d5 by move 2. 1.e4 e5, 1.e4 e6 2.d4 d5, 1.e4 d5, 1.e4 c6 2.d4 d5, 1.e4 d6 2.d4 e5, 1.d4 d5, etc.

By the time I finished those 3 books, in about 3 to 4 months time, I had felt comfortable about one structure over the others. In a random 5 minute blitz game, after 1.e4 e6 2 d4 d5, I asked "Does this opening have a name?". I was told Yes, it is the French Defense. My first opening book was Winning With the French by Uhlmann, and that I why I have known that opening for over 29 years.

Let the opening come to you, like a baby becoming left or right handed. Forcing the issue will simply make matters worse for you.

So, 1600.from 700 is an almost impossible task. How can I break through 700s in the first place? I mean, numerous times I have heard in this rating range we hang pieces a lot. Well, we don't do on purpose, do we? I don't understand when I hang piece untill that is captured. Many times I don't understand why this move is good, that move is bad. Would any of you suggest any realistic process to get past 900? Highest rating of my lifetime is 826.

Look at this. That is probably the least recommended opening from black . Still I didn't manage to win it. There must be something fundamentally wrong.

You ended in a draw, and not because of the opening, but end game play. The game goes through phases, the opening, the middle game, and the ending. You can come out of the opening well and still lose a game, but that doesn't mean the problem was with your opening.
Early on you don't need to study openings all that much, rather, focus on a few key ideas. At 700, just get either your e or d pawn to the centre. If you're with the White pieces, that just means start with either e4 or d4. I suspect your games with the Black pieces mostly start that way, and the few that don't "stand out". Personally, ignore the stand out games for now, they are anomalies. You can deal with them later, but for now, you need to gain useful experience. By that I mean, you need to try and repeat the opening few moves in as many games as you can so you can work out what works for you and what doesn't, otherwise chess is just a maelstrom of new moves to figure out every game!
So, as White, choose either e4 or d4 as your starting move and then, if you're not already sure, keep track of what your opponents usually play. Are you playing e4 and most often getting e5? Ok, then look at the e4 e5 openings and pick one. Personally, I would suggest the Scotch, (1. e4 e5 2. Nf3 Nc6 3. d4 ed 4. Nxd4 ...) as there aren't as many lines you need to learn, the starting few moves are pretty common (note if Black doesn't play 3. ... ed but defends their d5 pawn with d6, then take their e5 pawn, trade Queens as Black looses the right to castle and you get castled, bring your rooks to the middle and just try to attack the Black King which has been hobbled a bit in terms of getting safe - if you don't pounce, you'll give them time to get to safety). The Scotch is pretty straight forward in its ideas, there aren't a lot of lines to learn, and you will often end up in familiar middle game positions. Eventually, as your middle game play improves (which it will if you get to practice it from similar positions), you can start to think about growing beyond the Scotch Game. But, as Kasparov showed, it's a good secondary weapon to have at your disposal.

As Black, do you find you face e4 or d4 more often? If e4, then pick something that makes White "play your game" - so 1. ... e5 or the French, or Caro Kan, or Sicilian. The last of those, though, has too many options for White, in my opinion, so I would avoid it. If you choose 1. ... e5, you'll probably face 2. Nf6 most, so look at Petroff's as it gets you into your "game" where you can focus on the lines that follow. Similarly if you choose the French (1. ... e6) or the Caro (1. ... c6). At 700, you only need to know about 4 moves, as your opponent is likely to deviate from "book", but the goal is to try and start the game by reaching similar positions.
If you get a lot of 1. d4 ... games as Black, then do something similar.
The idea is to find something to deal with the biggest "chunk" of the games you play. You can't deal with everything at once with a "prepared response", so that's where opening principals come in. There are lots of resources on the web that can teach you those far better than a post here can do, so take a bit of time to watch and digest those ideas.
While many will say "don't learn openings", what they mean is don't spend a lot of time memorizing opening lines 10 or 15 moves deep. However, you do want to try and start your games in a similar way so you reach similar positions and then solve the problems your opponents pose. As you improve, you will play stronger opponents, who will in turn start to pose similar problems early in the game, which you meet by then, and only then, learning a couple more moves of your "opening", and the process continues until suddenly you realise you've learned some lines out to move 15 or so, but only because now you are facing people where you needed that.
And yes, at some point you'll realise you can't only play the XXX (i.e. the Scotch), and you probably will want some variety as well. But by then, you'll have a better idea of what options look appealing to you and what depth you need to learn a new opening for your level will be obvious.
Chess is a great game. While the openings are fun to study, and provide a sense of security, just claim or at least fight for the centre of the board (move central pawn out), get your minor pieces out in the order of Knight then Bishop (usually), get your King safe (castled), and for the love of friends and family, don't ignore your rooks and leave them sitting on the side of the board but bring them in to a square on a file that is going to open up at some point!
Anyway, the far shorter version is, just have fun. If you want to study to improve, then your time is better spent on tactics and positional ideas of the middle game, and also on end-game play. Openings, just pick something common, but limits your opponents options (meaning, at move 4 you don't see things like "and now your opponent has the following 15 possible decent moves ...." - trust me, you don't need that sort of aggravation!

That game's result had nothing to do with the opening. And your opponent played their not recommended opening pretty well. Even if the opponent doesn't play by the book, you still have to play better than them to win. That's the only way. There are no shortcuts. If you repeat moves in a winning position, you'll get a draw, and it's not anybody else's fault.

When you are a rook and a knight up, and end the game with triple repetition while you haven't consumed half of your available time, it's clear that the opening is the very last thing to blame.

The final position (before 53.Nc4??) is mate in three moves, at most- and a very easy one.

Nnnx

Okay, sorry if I blamed opening. My intention was to put some light on the fact that I am not being able to manage wins even in a mate in 2, because it's not something like a back rank mate. What can I do about that?

Get familiar with other checkmate patterns beyond just back rank mate. You can find info on chess.com or the web (things like dovetail mate Anastasia's mate, Arabian mate, etc). And puzzles are a good way to learn how to recognize when you should consider trying to maneuver the position towards one of those.