EDIT: It was really remiss of me not to mention http://arxiv.org/abs/math-ph/0211040 by Patrik Ferrari and Herbert Spohn, where there are very nice results concerning scalings when the distance between the two directions varies with $n$. The case $\beta=\alpha+O(n^{-1/3})$ is the one where a very nice scaling picture should emerge; maybe http://arxiv.org/abs/1103.3422 (Corwin and Quastel) on the "Airy sheet" is the beginning of this.
Hi Nathanael. The answer to Q1 will be $O(1)$. For concreteness consider the model with i.i.d. exponential weights at each site (the case with geometric weights at each site should also be fine but the geodesics are not unique so it's slightly messier). The path $\gamma_n(\alpha)$ converges almost surely to a path $\gamma(\alpha)$ as $n\to\infty$ (a "semi-infinite geodesic in direction $\alpha$"). So the question reduces to looking at the intersection of $\gamma(\alpha)$ and $\gamma(\beta)$. (The same behaviour should be seen across a wide class of models but showing that rigorously could be a challenge).
Coupier ( http://arxiv.org/abs/1104.1321 ) is a starting point for results and references concerning existence and uniqueness of such semi-infinite geodesics.
Actually one could certainly say some specific things about the law of the intersection of the two semi-infinite geodesics (in an exactly solvable model like the i.i.d. exponential weight case). The joint law of the geodesics in two different directions from the same point is very closely related to the equilibrium of an appropriately defined two-type particle system. In this case the relevant particle system would be a certain type of process like the Hammersley process, but in discrete time and space and with "fluid" rather than discrete particles. Eric Cator and I have done some work relating joint laws of geodesics to two-type equilibria in this way recently - it is not written up yet but I would be delighted to discuss. The closest thing on paper might be Eric's article with Leandro Pimentel at http://arxiv.org/abs/0901.2450 . The kind of calculations one could do concerning the two-type equilibria might resemble those done for the TASEP by Amir, Angel and Valko in http://arxiv.org/abs/0811.3706 (which make use of a queueing-type representation for the multi-type equilibrium of the kind that Pablo Ferrari and I developed in http://arxiv.org/abs/math/0501291 ).