I have answers to question 1,2 and 3.
1) True. This is essentially a restatement of existence and uniqueness of Teichmuller maps between two points in Teichmuller space. The Teichmuller map between two marked Riemann surfaces $X$ and $Y$ gives a quadratic differential on $X$. Stretching along the vertical foliation and contracting along the horizontal one gives the Teichmuller geodesic between $X$ and $Y$.
2) False. There are Teichmuller geodesics which do not have a well-defined limit at infinity in Thurston's compactification. An example is given in the reference by ThiKu. One can nonetheless study their limit sets in the boundary.
3)False. The quadratic differential associated to a Teichmuller geodesic defines a pair of measured foliations $F_1$ and $F_2$ which together fill the surface, meaning that any simple closed curve has positive intersection number with either $F_1$ or $F_2$. It is easy to build a pair of simple closed curve on a genus two surface which do not fill it.
I will add that there is another difference between geodesics in the hyperbolic plane and in a high genus Teichmuller space. In the latter case, there are examples of distinct Teichmuller rays starting at the same point $X$ which converge to the same point in Thurston's compactification.
Examples were first constructed by Masur in the paper "Two boundaries for Teichmuller space".
I hope that it helps!