This is an example of an absorbing Markov chain, where the states are Player 1 wins, Player 2 wins, tie, and then any combination of Player 1 having 0/1/2 points and Player 2 having 0/1/2 points. We can construct a Markov matrix where the jth entry in ith row is the probability of going from state i to state j after one two-die roll by both players. We can determine these probabilities pretty easily, since the player scores are independent of each other - e.g. the probability of going from P1 having 1 point and P2 having 0 points to P1 having 2 points and P2 having 2 points is simply the product of the probabilities of P1 getting 1 point and P2 getting 2 points.
Once we have the matrix, we can apply the formulas for Markov matrices to answer your questions - by raising the Markov matrix to an infinite power (or approximating the result of this by raising it to a high power) we obtain a matrix which gives us the expected distributions of the final state given a starting state, and we can also obtain the expected number of steps needed to reach an absorbing state (in this case, either player reaching 3 points) from a starting state by summing the state's corresponding row of the fundamental matrix. The exact formulas can be found on the wiki pages for Markov chains and absorbing Markov chains.
Here's a download link to the Excel sheet I used to actually runs these calculations - if you change the value in Q1 to the probability of rolling a 6 on any one die in this example, it will generate for you the probabilities of either player winning or a tie, as well as the expected number of rolls needed to end the game. For p = 1/6, either player has a 46.6% chance of winning, and there is a ~6.8% chance of a tie, and games on average will take ~6.69 rolls. For p = 1/3 the probability of a win is ~42.4%, of a tie is ~15.2%, and the expected number of rolls is ~3.61.