Is it true that PSD-cone always convex? (If not, please provide an example). If this is the case, then set of PSD matrices always convex or it could happen that such set might not be convex for some combination of PSD matrices?
As far as I have understood, the set of PSD matrices is always convex. Am I right?
Yes, the set of positive semi-definite matrices is a convex set.
It is rather easy to understand intuitively as the set is all $X$ such that $v^TXv\geq 0 ~\forall v\in R^{n}$, which effectively is an intersection of infinitely many linear inequalities.