What is Caruana trying to prove?

The rating spots were decided based on old rating lists, so Aronian and Kramnik got those long ago, or maybe that should be Aronian and Karjakin after Kramnik won the World Cup. Caruana is not mathematically eliminated from second place in the Grand Prix series though. He just needs to be sole first here, and if he beats Gelfand with white today he will probably be shared first with three rounds to go.

I was playing ten minute game and my opponent had a completely lost position. He would not move, I disconnected and reconnected thinking I could trick him into moving and viola he moved when he saw I disconnected. THAT is poor sportsmanship. Playing on for a few moves after a blunder is not poor sportsmanship in my opinion. Especially if he resigned a bit later.

He was probably disgusted with himself as well as flustered and when he regained composure saw that he should resign. A loss like that OTB is a gutting feeling and can shake you up.

I don't think it was a "fingerslip." Caruana spent a lot of time on that move (this was said in his post-game commentary). He couldn't explain why he made that move. He said that he remembered a game where fxg6 was played. But in the end he just played a bad move like all humans do.

What you didn't see, was the poker game later. Caruana was playing Grischuk, bluffs with nothing, and beats Alexander. Nakumuru, who was watching the game, tells Grischuk:

"Nothin'. A handful of nothin'. You stupid mullet head. He beat you with nothin'. Just like today when he kept comin' back at me - with nothin'."

Caruana tells Nakumuru:

"Yeah, well, sometime's nothing is a real cool hand."

Nakamuru joins the game and says:

"Let me sit next to my boy--Cool Hand Caruana..."

Yes Pfren, but I don't get why not black takes the rook in move 25 or 26. Also why it's 1-0 after whites move 27 when black just can take on d2 and win? 

Caruana played on in order to avoid the double embarrassment of losing badly and having given Nakamura a miniature. He played on past move 30 to make the score look respectable. Years from now, when people are looking at databases of the games between Naka and Caruana, this game will not stand out from the others.

Perhaps resignation was better, but playing on past move 30 is fairly common among GMs who blunder early.

Didn't Nakamura drag out a game against Aronian earlier this year?

Probably that he's really good ^_^