tl;dr: Normal low-rated shenanigans. Both sides did not play well. Black played horribly in the beginning, white didn't know how to win and imploded. Plus the time control is too short for you maybe.
Actual content:
3. cxd5 - the idea is not uncommon (look up the Exchange QGD) but the execution is. The pawn is not normally taken on move 3; black has the option of say 3. cxd5 exd5 4. Nc3 Bf5, with an easier time than usual.
6...Qd6 is perfectly reasonable (although 6...Nf6 or something developing is slightly better.) The queen may be safe momentarily, but leaving it there would be risky and perhaps give chances for later tactics by white. Also, you expected 6...Bxc3+, but that's not a good move; black does not want to let white have the bishop pair in a favourable position, and white's centre after 7. bxc3 would be strengthened. Black should only consider ...Bxc3 if white uses a move to force it (e.g. a3) or if it leads to some positional/tactical gains (not in the current position.)
Consider 7. Bd3/Be2 and short castling soon. It's usually a good idea to get the king safe quickly, and here the wing is safer than the centre.
9. bxc3 feels more logical. More pawns in the centre.
10...Nxf2? - Probably a highly optimistic beginner move. Some people believe sacrifices will always lead to a successful attack (protip: nope it don't.) The only engines that would play such a move are really weak ones anyway. And why would you cheat with those?
13...Re8 - Nothing suspicious here. In a normal chess game your opponent is supposed to anticipate your ideas, not sit there and get steamrolled.
17. Nxe5? - Just take the rook with 17. Qxa8. Instead you let black have the chance for 17...Bb7 saving the rook.
19. Qf3 right away was slightly better - black's bishop was attacked on d7 and needs to waste a tempo to move away, unlike on e6.
You are winning severely at this point. Now what went wrong? Diagnosis: You don't know how to win while up material.
28. g3? is bad. Simply 28. Kg1 and black has nothing on your extra 2 rooks and safe king. In fact 28. g3? hung the rook (28...Qxh2+ 29. Kf3 Bd5+; 29. Kf1 Qh1+ and ...Qxa1 or xc6; 29. Ke1 Qh1+ and ...Qxc6), although white's still winning.
Then you missed 33. Qe7+ Kg6 34. Qd6+ winning the bishop, or even better 34. Qc6+ forcing a queen trade (you're up a rook already, better to have queens off the board than win another bishop.) Ditto for missing 34. Qe7+ Kg6 35. Qc6+ with the same idea.
42...Bc6+? 43. Rxc6 is a free bishop. 51. Rc6+?? is suicide.
Points to learn:
How to win a won position? King safety. If you're up material without compensation, especially ridiculous amounts like a whole rook for nothing, you know the win is practically (but not yet) in your hand. What better way to throw away the win than let your opponent get perpetual check, or tactics winning back material, or perhaps even checkmate you? To prevent this, king safety should be the first thing in mind. Keep your king safe, use your huge material lead to trade down to an obviously winning (and safe) endgame or just crush your opponent slowly. 28. Kg1 is the most blatant example of this approach that you missed.
Time controls. You seem to have got into severe time trouble near the end, and got spooked by your opponent's fast moves. To remedy the first problem, play longer games where possible. 10|0 (this game's time control) is not a lot of time for a new player. Try longer games; I used to play lots of 25|0 when starting out (back in the day when increments didn't exist.) On this site, 30|0 is a good control; 15|10 is a tad fast for new players.
To remedy the second problem (and simultaneously allay your suspicions), don't pay heed to your opponent's clock or speed of his moves (except to notice when he flags, but this site does it for you automatically in Live anyway.) Just focus on playing the best moves you can, while managing your time wisely (always leave yourself enugh time to finish winning the game.) In this game, black was dead lost so playing fast moves wouldn't hurt him either; it just so happened that the obvious moves like checks were good for black (and you let it happen.)
This player started out and seemed to struggle - made some mistakes early and took a lot of time. Then, midgame, suddenly came to life, playing perfectly and every move was instantaneous - should I be suspicious? I don't think I made any huge mistakes, but even with a massive material advantage, just felt like I didn't even have a chance....like he anticipated my every move and was five moves ahead.
Just curious what people think - I certainly made some mistakes here, and decided to go for material (namely his two rooks and the knight he traded for a pawn) at the expense of positioning, which he ultimately was able to use to his advantage. I welcome any insights and/or critiques.