What role does psychology have in playing a game of chess?  Is the best move the one that will stand up under endless post-tournament analysis, or is it the one that makes the game most difficult for your opponent?

How legitimate is "psyching out" one's opponent?  Nimzovitch was the target of such psyching out in the famous cigar anecdote with Steinitz (Steinitz is supposed to have been waving a cigar around at a tournament in which smoking was forbidden; Nimzovitch complained, saying "The threat is stronger than the execution"), which is a good example regardless of how apochryphal the anecdote itself may be.  Alekhine's reported joke at Bogulyobov's expense (dying and meeting St. Peter at the pearly gates, A. is refused admission on the grounds that he is a chess player; he sees B. and objects, whereupon St. P replies, "He only thinks he's a chess player") was very possibly another form of intimidation.  Even playing an opening that you know your opponent is scared of, whether it's the best line or not, can give you a psychological edge.

And last: How much do we lose in online chess through not being present together with our opponent and having the opportunity to apply the psychological screws?  Or how much do we gain?

The psycology factor is almost non existant in online-correspondence chess. But in a real case scenerio like a tournament it is HUGE. As a matter of fact its one of the main reasons people study other´s games before they play with them...you determine your opponents style of play and make him/her go against the grain. Its very hard to play a closed game when you are very tactical or viceversa... it can severely limit the performance of even a highly skilled opponent to make him fight on different terms that he is used to.

About the moves... for all practical pourposes making a dubious move on internet chess is just bad. No matter how uncomftareble your opponent may be in a given position he has time to work it out-theres no rush, so it would be a bad idea to play strange moves on internet correspondence chess. On the other hand making moves to annoy your opponent and unnerve him/her is an excellent tactic in timed chess, it is often not the best move that wins the chess game, but the move that your opponent is irritated with and his state of mind will not allow him to find the correct defence.

I agree with KingsMove that the psychological element of the game is greatly diminished in online correspondence chess.  But what he says about playing to an opponent's weakness or discomfort applies here as well.  In fact, the format of chess.com encourages this.  Instead of walking into a tournament hall and watching your prospective opponents over the board, we can look up a prospective opponent's games archive and see what s/he does and does not like to play.  It's true, however, that that an unexpected move will probably not have the influence online that it does otb.

Online, a player cannot roll a cigar between his fingers or take out his Bowie knife to clean his teeth while his innocent opponent is trying to consider how the move just played relates to the Greater Truths of Chess.  And I'm sure we're all better off at some point that chess.com does not offer a cam2cam component.  But just as we are protected from these things, we also cannot read our opponent online.

There are, however, annoying practices players can adopt when a game is not going well that can color one's whole attitude towards a game and tempt one into unsound moves.  However, I shall not go into those things here, as they have lately been well rehearsed in other forum threads.

another great thing is the "disable chat" option - if opp tries to distract you by making you angry, just disable chat, take a break to cool down - may be play quickchess or a 15 0 game here or on fics or something - if you have enough time left skip *moving* in that game for a couple of days.

[unless my reply is obvious (or obviously routine) i anyway download the pgn and view it in xboard, and put the pos. on a handheld pocket chessboard [good magnetic so the pieces dont fall down :) ], apart from *writing down* my moves. I log off, analyse and decide my move - often on a big chessboard on which the pocket chessboard pos. is copied by hand - doublechecking the fen.

After i decide my move, i make it the next time i log in. 

That way I play the board not the player :)