Yup. Those numbers (indicating points for win, loss, or draw) are mathematically calculated based on your rating compared to your opponent's rating. Chess.com uses glicko rating system (super close to elo in calculation except the k-factor in the formula).
Becoming a chess grandmaster (or even a titled player) is usually much more than just winning a lot of games though. For instance, to become a grandmaster (at least via FIDE), you typically need to meet specified norms. Based on which title you are going for (GM, IM, FM etc.), there are different norms to be met; many of them are playing x number of games against an opponent of that title you are seeking. A grandmaster needs to play some games against players who are already grandmasters and so on; you can't become a grandmaster solely by playing lots of games (especially versus weaker players) and trying to build your rating. 
1. At the beginning of a game, it will say who you are against and a bunch of random numbers next to the words "win, draw, lose". Newsflash! They aren't random numbers! Those numbers explain that based on the turnout of the game, you will either lose or gain a number of points to your chess rating. In order to become a Grand Master, you will need to win lots of games.
2. You can't chat in tournament chats until later on.