In theory, if he does it enough times in the same month, then yes. Ratings are only updated once a month, and in the meantime, no matter what the live ratings of the two players are, the rating change in the case of a draw (or a win/loss, for that matter) is entirely based on the two players' last official ratings. Currently Carlsen loses 1.1 points for a draw, and Karjakin gains that many points, so if they were to draw 50 games before the official ratings are updated, then yes, Karjakin would surpass Carlsen.
Can Karjakin surpass Carlsen ELO rating just drawing with him enough times?

However, starting from Game 0 and their initial ratings, 2853 and 2772, the best way for Karjakin to overtake Carlsen would be to win every game, therefore 7 games.
The difference is 81 points. With every lost game, Magnus lose 6.1 and Sergey gains as much. This means MC would drop to 2810 and SK jump to 2815. Well...

Karjakin has gained 13.8 points so far and his live rating stands at 2785.8, Carlsen losing (obviously) the same amount and has a live rating of 2839.2.
Karjakin is expected to score 0.39 per game, so a win would give him 0.61 (1-0.39) multiplied by his k factor of 10 = 6.1 elo per win (as MSC157 points out)

Caruana is threatening Carlsen's first position in the ranking.
Well... it's not su much Caruana threatening Carlsen's position as Carlsen threatening his own position with his bad play.
But regardless, even if Karjakin wins the championship, I don't think Carlsen is going to drop below Caruana, and I think he's going to continue to be #1 in terms of rating for the forseeable future. Caruana's current rating is not sustainable - if the tournament he just played in had been rated, he would have dropped quite a bit. Carlsen, on the other hand, I think is going to bounce back to 2850 soon regardless of the WCC result.

The answer to the initial question is of course NO. You cannot surpass somebody by just making draws. You only can have the exact same rating after many many draws.

The answer to the initial question is of course NO. You cannot surpass somebody by just making draws. You only can have the exact same rating after many many draws.
Except that official ratings aren't updated after every game, so if you draw enough games in the interval between two updates you can in theory surpass a higher-rated player.

The answer to the initial question is of course NO. You cannot surpass somebody by just making draws. You only can have the exact same rating after many many draws.
So, Karjakin vs Carlsen: how many draws?
7 draws, 1 win

Yes. If Carlsen then loses to other players.
If they are only playing each other forever and draw every time their ratings will eventually stay the same (as the same).

The key is that official ratings are only updated once a month, and between updates the rating adjustment for both players in the case of a draw is determined by their last official ratings, not their live ratings. (Take the last round of the 2016 Tal Memorial for instance, where Nepo drew against Gelfand. At the time, Neop's live rating was in the 2760s, and Gelfand's was in the 2720s, but neither's rating changed as a resut of the draw, because their last official ratings at the time were both in the 2740s.) In the case of Carlsen and Karjakin right now, Karjakin gains/ Carlsen loses 1.1 points for a draw regardless of what the live ratings of the two players are. Right now, the live ratings of the two players are 2839 and 2785, respectively. If, for some reason, they were to play 30 more games by the end of this month and draw all 30, Carlsen's rating would drop by 33 to 2806, and Karjakin's would rise by 33 to 2818. So just by drawing against Carlsen, Karjakin could in theory pass Magnus in rating, without either player having to play against anyone else.

The question is whether Carlsen vwill be the same Carlsen if he loses the title. Losing the game was a serious psychological blow for the guy to the point that he reacted like a girl on the press conference. And I am rooting for Carlsen by the way.

The question is whether Carlsen vwill be the same Carlsen if he loses the title. Losing the game was a serious psychological blow for the guy to the point that he reacted like a girl on the press conference. And I am rooting for Carlsen by the way.
I wouldn't be too worried about Carlsen, and I'm not rooting for him. His reaction at the press conference is how he always reacts to losing - it's a well-documented habit. If Carlsen does lose the WCC - and I think he probably will - I would expect him to bounce back to 2850+ in a few weeks, and go back to being the same old Magnus, except now with a vengeance to win back his title.
But I already said all of that in the other thread, and you probably saw my comment there.
Question is in the title