I won against a 1800+ player! Did he play seriously?

This is the game (I was playing white):

It was a casual 10/20 game, so I'm afraid he didn't take me seriously... For example, I find his move 19...Rf8 is suspictious, because I blundered giving away my knight, and he could take it after a rook exchange, but he didn't. And his move 23...Rd4 is weird too. What do you think?

PD: By the way, although I appear to be rated 1636, it's false because I haven't played/lost enough games there. I don't think I play better than a 1400 player (as it is shown here).

What they call "casual" means that it doesn't affect ratings. It was an unrated game. Based on that alone, I would guess he did not play "seriously".  At any rate, you played well enough to win. Enjoy it.

That lichess link to the game and the analysis of it, your ratings, etc. is fascinating. Many people would love the move by move analysis it gives.

As far as your rating, it shows that you have played 12 games, so your rating is fairly legitimate, for that server. It's not a USCF or a FIDE rating or a chess.com rating. Ratings are not absolute. They only compare the people in the group playing each other.  You can't compare your lichess rating to your chess.com rating and feel badly about it.  It's only a lichess.com rating based on 12 games. If you want it to become more legitimate,  play rated games.

Rating is provisional for the first 25 games I believe.

Huh, it is surprising to me as well... this does not seem like 1885 chess on the part of black. That said, maybe the individual was tired, or not focused because the game was casual (and perhaps distracted by other priorities: i.e. he/she should have not been playing chess at that moment). One last thought, I have heard that some individuals inflate their ratings by using chess engines from time-to-time. All that said, some times people under-estimate their opponent's potential skill level by focusing too heavily on their rating & having the preconceived idea that it will be an easy win. This is why it is better to treat all opponents with the same respect & play in the moment.

Should white have exchange the white bishop for the knight? I've heard that white should keep the white bishop for as long as he can. 

Of course I am not the best person to answer your question about the white bishop - knight exchange, that being said, nothing in chess is absolute. If one gets too caught up in the structure of how one is suppose to play, then many opportunities will be missed.

In this case, I think white was aiming to maintain control of the game & to double up Black's pawns which makes them less useful (and with a more-or-less equal exchange to boot).

That being said, I have heard from Josh Waitzkin that if one can maintain the "Tension" that one will be better off (meaning to delay any exchange for long as possible until the right moment). This of course is a skill that needs experience for developing. Good luck!