Bobby Fischer teaches chess is very much at the absolute beginner level. Game 1 the Ne5 move was way premature, but you explain that.
This guy wanted to see if he could master chess in 30 days. He improved, maybe going from beginner to 1000 or 1100 in 30 days. On day 30 he has a rematch with a 1500 player that does not go well. I had much to say about that game in the comments section of the video.
To really master chess you might want to learn 1. d4. The book "Logical Chess Move by Move" is useful in this regard.
The en-passant rule has to do with the fact that pawns used to only move one square from the start. Moving two squares was a way to speed up the game (and so is castling), so enemy pawns can normally capture your pawn if you move up one square and this is important for allowing pawns to blockade other pawns. En-passant allows a capture as if the pawn had only moved up one square, which maintains the ability to blockade.
My experience is that you are unlikely to reach a memorized opening past move 11. There are some exceptions where certain openings, like the Dragon, don't really get started until around move 10. Everything up to that point frequently follows the same sequence.
In the last game, I don't like 5. Bd3. It is not a great spot for the Bishop, which is serving the purpose of a pawn. The move b2-b4 creates weakness on the C file with a backward pawn on c2. Here your pawn is well guarded but there is a big hole on c3 creating a weak spot that your opponent should try to occupy. Your pieces (bishops and knights) are somewhat passive and in defensive positions, and eventually you will want to get them to better spots. (Grandmasters really like to have active pieces.) Ouch, you got skewered with Ba6! The game is technically lost at that point. The Bb6 response is probably bad. I think that he should respond with Qb7 and try to pin your b6 bishop. He didn't have to capture the queen immediately. Instead of Nb3, the knight taking on e4 makes more sense to me because material is important. When you played Nf6+ (check) you had two attackers on the square and he had two defenders on the square, but you moved one of your attackers to the square meaning that he was going to win the piece. You really need more attackers than defenders on that square to make that move. After he captured the knight it is better not to trade your bishop because you should not trade when you are down material, because your percentage of the total material becomes worse. Likewise, you should not have traded your last piece, but the game is pretty much over by that point. (Overall, you were too anxious to trade pieces. Many beginners do this, but I only trade when I think that it is to my advantage.)
John Coffey (rated 2016)
My free online chess lessons: http://www.entertainmentjourney.com/index1.htm
Pattern recognition is a huge factor in this game, and on my website I stress doing tactical exercises to improve pattern recognition.
P.S. Your hypothetical example of where trading pawns might be favorable to Black is really bad because after 1. e4 g6 2. d3?! f5? White can play 3. exf5 and then if 3... gxf5???? 4. Qh5# (checkmate), which is a variation on "Fools Mate."