Won a piece, played poorly, and then lost. What went wrong?

I'm white. The time controls are 15+10. Here's the game. lichess.org/nh5Y4wnE

After winning a piece by move 9, I was pretty sure I was going to win, but I had trouble finding good plans, got into time trouble, and somehow ended up losing. Could any of you take a look at what went wrong?

Here are some specific questions I have:
- On move 14, I spent some time trying to find a plan. I didn't like how black had many pawns in the center, so I decided on a plan involving the moves Ne2, c4, b3, and Bb2 in order to gain some control of the center. Is this plan good or not?
- I thought that 20.Nc3 was bad, given that the knight moved again later anyway. After the game, I thought that the best move was 20.Rd2, possibly continuing with 20...f5 21.g3 e4 22.Bg2, when the e2 knight is well placed to take advantage of the pawn structure. However, 20.Nc3 turned out to be the computer's second choice, so it's actually pretty good. Why is this the case?
- During moves 21-24, the computer is suggesting plans involving b4. Why? I thought I had to deal with black's central pawns somehow.
- The computer suggests 25.g3 and 27.g3. I considered playing g3 at some point around those moves, but I rejected this plan since making weaknesses when the opponent might have an attack on my king seemed terrible. Why is it a good idea?
- I knew that my 24.Ne2 and 25.Nd4 plan seemed dubious, but I couldn't think of any other plan. I'm having trouble finding good squares for my pieces. What kind of setup should I have done?
- During the game, by move 27, I was already preferring black, and I thought that if I did nothing, black would win with some kingside attack. So I played 27.Rd5, giving up an exchange, which I thought was the only was to keep the advantage. The computer doesn't like this, suggesting (again!) 27.b4 or 27.g3, when white is still winning. Why is black not having sufficient compensation for the piece? And what's up with these b4 and g3 ideas?
- I couldn't play 30.Bxa6 due to Ra8 and Rxa2. I missed the idea of playing 30.a4, forcing a5 since black cannot defend both e4 and a6. But what's up with 30.b4 again? I thought I'm supposed to keep the position closed so black's rooks cannot get active. My idea of 30.f3 was to open up the d3 and e4 squares for my minor pieces, but apparently it was bad. I knew that putting pawns on light squares is bad for my bishop, but I wasn't sure how I should have coordinated my pieces.
- Was my 31.Na3, 32.Nb1, 34.Nc3 maneuver a good idea?
- Why is 35.Bd3 an inaccuracy? Why is the position after this move equal?
- After I let the computer think about the positions after 47.Nxb4 but before 49.a4 for a while, the computer says that it was OK, and the position was equal. Why is black not already having an advantage due to the king becoming more active?
- What's wrong about 49.a4? I thought I'm supposed to push a passed pawn.

Please ignore my moves after 49.a4, since I started making mistakes due to time trouble. Also, if there's anything else you noticed, please tell me.

Very humble opinion, I'm absolutely no expert, but after 21. ... e4, your light square bishop is useless, if you want to save it you're going to cramp your position. On the other hand, black has an excellent center with the pawn e4 and f5. So, would it be a bad trade 22. Bxe4 fxe4 23. Rxe4? I think not.
After that, you can immediately put d6 under pressure, and if your opponent push d6 - d5, trade it with your c pawn, black is down a pawn and has two isolated pawns.

15: My plan would be to play Bf4 and try to pressure the castled king. With B on b7, black seems to have limited defenses.
20: I prefer g3 with a blockade on the dark squares. Nc3 is only good for restraining d5, which Rd2 seems better.
21: One of my recurring plans would be to restrict B on b7. b4 and c5 seem to blockade that piece further. Of course, the computer may be thinking that a4 and b5 to get a passed pawn is a good plan.
23: Last chance for c5 and blocking in black's bishop.
25: All of your pieces are passive. Is Qa1 an active try?
39: Rule-of-thumb: black's rooks are redundant, and you shouldn't trade rooks--not to mention open the position.
47: I think the N on c6 is a monster--restraining both black's pieces. My plan would be Kd3-Bd1-Kc4xb4 with the a-pawn winning the game.
49: a4 loosens the bishop. Just shifting the N between e1 and d3 would be a fortress.

As I'm only a 2100, my opinions shouldn't be any better than yours, especially since I spent much less time on the game.

Ive noticed during the middlegame you were struggling to create a plan but you HAD to go b4 and do a pawn break, instead you made a lot of piece moves which gave black the initiative and a lot of free pawn moves to cramp your position, and meanwhile your pieces just ended up worse than before (moves 20-32)

Yes you were up a knight for a pawn, but you had trouble finding a good square for your knight.

25 Nd4 was no good, as 25...c5 pushes it back to c2 and that also opens the diagonal for Bb7. You see you havel one more knight, but at c2 it does not let its presence felt. The presence of the extra pawn, however, is clearly noticeable as it fuels his attack. So indeed 25 g3 was better, but it is not a move one likes to play. No harm is done by 25 Nd4, but it makes it harder for you and you will voluntarily restitute an exchange with 27 Rd5 to stop the activated Bb7. You were still winning.

35 Ne4 would have been strong: you finally had a good square for your knight. Why did you retreat your knight to e2? You have now knight and bishop for rook and pawn, so your advantage depends on finding good squares for your pieces. Now your advantage has evaporated and the game is even.

47 Nxd4 allows his rook into your rear. This is the tipping point where black gets the advantage.

His 52...Rxa4 lets you off the hook. He could bring his king into the battle with 52...Kf6 and ...Ke5.

64 g4 was a losing blunder of course. You are about to lose pawn g2, but allowing him a protected passed pawn on g3 is far worse.

b4 to justify the position of the rook on b2 and to open up some lines for your pieces assuming you get to play b5 afterward and trade pawns.

I'm too lazy to analyze everything else.

Reply to #2 (@Topobanana):

I didn't even consider giving back material. The computer doesn't like it, but the resulting position looks so much easier to play. My rooks suddenly become well-placed too. I would probably prefer taking with the knight though.

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Reply to #3 (@jonesmh):

Move 15: I didn't play 15.Bf4 because I thought black would play 15...e5, and I had to move the bishop again. The computer says that I can play 16.Bxe5, though, and the tactics work out for white. But then, black can still play 15...d6 and save e5 for later. I'm not sure why I should play 15.Bf4.

Move 20: I have no idea why I didn't play 20.g3.

Moves 21-23: For some reason the idea of blocking the bishop or generating counterplay on the queenside didn't even occur to me. I guess I was too fixated on center control.

Move 25: I'm not sure what 25.Qa1 achieves after black just moves the queen away.

Move 39: I thought that exchanging rooks with 39.Rxb4 allows my knight to go to c6, and with only one rook, it would be harder for black to defend the weak pawns.

Move 47: 47.Nxb4 was a mistake made in time trouble.

Move 49: I thought I had to do something since black's king is coming. However, it turned out that black can't do anything even if the king comes to b2.

In this game, my performance is probably much, much lower than my rating, so don't worry about whether you can give me advise.

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Reply to #4 (@arasvard1) and #6 (@MyPunIsTwoWeek):

For some reason I got into full defensive mode or something. Didn't think of using pawn breaks for counterplay at all. I was probably too fixated on center control.

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Reply to #5 (@tpr):

When I played 25.Nd4, I thought that c5 would weaken black's pawn structure. And for some reason I thought that c2 was a good square to stop black's pawns, and not block the rooks. But I soon realized that black's bishop could become a problem, and that my knight needs to be repositioned. My main idea of the exchange sacrifice wasn't eliminating the bishop, though, but to force a queen exchange.

I couldn't find a plan of what to do after 35.Ne4. The only way I could think of to proceed is to target weak pawns, which is what I did.

47.Nxb4 was a mistake made in time trouble.

The reason I played 64.g4 was that (in time trouble) I thought I was losing at least two pawns, so I thought I was losing anyway. I didn't see the Ng5-h7-f8-g6 idea as the computer pointed out.

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Thanks for the comments.

Even if the engine doesn't like giving back material, you have to play on the basis of what you can do. Then, knight or bishop, is just a matter of taste.

Here is my analysis of where you went particularly wrong:

8... Qa5 you just went up a whole piece without compensation.

On either move 20 or 21 (preferably 20) you would have been wise to play g3. This increases your king safety and allows you to morph it into a bishop fianchetto position. g2 is a better square for the bishop than f1. You are still winning without doing this, but not by as much.

Again on move 24 g3 is crying out to be played.

Move 27 with Rd5 I can sense your fear you are sacking the exchange for safety.

After move 29 things are almost equal but personally I prefer the 2 minors vs the rook.

30 f3 allows e3 and that pawn is now messing things up for you

Moves 34,35, and 36 your knight is crying out "please let me go to the lovely e4 outpost... ignored

39 2 rooks come off and to me this looks like a drawn position where I'd ever so slightly favor black because of that nagging e pawn but the engine says dead draw

49 a4? WHY!? That pawn is doing a GREAT job of making an outpost for your bishop. As long as it is there that bishop can not be taken and look at that a pawns prospects... Do you really honestly think that it can queen or threaten to queen? At this point with white I would hop my knight to e1/d3 for as long as possible and if that's not possible the bishop or king have waiting moves. Totally should be a draw here.

52 Nc2 is there a reason you let them have that a pawn for free? The bishop can block here and defend

63 Nxg5 is practically fatal. The rook goes b7 then g7 and wins your g pawn and gets really strong on g3. Better is Nf2. Your 2 pieces & king should be able to cooridante together to stop the black d pawn and at the same time hold on to the weak white g pawn while avoiding being checkmated. Dead draw

64 g4 makes no sense when the knight has h7... I mean I don't like your prior knight move but I hate this move. I would sooner resign than play that move because enpasant is going to wreck things.

A player who played with white pieces could play more actively. 14. Ne2? ... What is it? Is the snake creeping? No, this knight decided to stay away from the battle. 14. Na4! d6 15. c4 0-0 16. Be3! ... With the idea of using a weak square b6 in a black camp . And in case of movement d6-d5, then use a weak square c5.
The one who has the advantage is obliged to attack.