Minimum pumping length of finite language
Let $w \in L$ be the longest string of the finite language $L$. Assume a pumping length $p \leq |w|$. This would imply $w = xyz$ with $|y| \geq 1$ and $xy^2z \in L$. Can you see the contradiction?
Having ruled out pumping length $p \leq |w|$, the minimal possible value for $p$ is indeed $|w|+1$, for which as you noted the conditions of the pumping lemma are trivially true.