Problem 1. The general statement may be deduced by induction from the case $n=3$, so let $T_1$, $T_2$ and $T_3$ be three subtrees satisfying $T_i \cap T_j \neq \emptyset$. Let $x \in T_1 \cap T_2$, $y \in T_2 \cap T_3$ and $z \in T_3 \cap T_1$. The geodesics between $x,y,z$ define a tripod with $[x,y] \cap [y,z] \cap [z,x]= \{w\}$. By connectedness, $w \in T_1 \cap T_2 \cap T_3$ hence $T_1 \cap T_2 \cap T_3 \neq \emptyset$.
In fact, the same argument works for median graphs. More generaly, there is an analogue of Helly's theorem: A family of pairwise intersecting convex subsets has a non-empty intersection. (See Roller's dissertation for more information.)
Problem 2. I gave two solutions to this problem in the other post Actions of Finite Groups on Trees; one may be based on CAT(0) geometry, and the other uses Bass-Serre theory.