About the question of uniqueness of maximal compact subgroups, up to conjugation, I propose a Riemannian geometric approach:

Theorem (É. Cartan). A compact group of isometries of a nonpositively curved complete Riemannian manifold has a fixed point. 

Proof. (Rough sketch) Consider any orbit. It is compact. By convexity (the curvature of the ambient is nonpositive), its center of mass can be defined and it is plainly a fixed 
point. QED

Now one uses the fact that the symmetric space of non-compact type $G/K$ is nonpositively curved. If $H$ is a compact subgroup of $G$ then it has a fixed point $gK$ by the theorem, so $g^{-1}Hg$ fixes the basepoint $1K$ and hence is contained in $K$.