For every $u>0$, there exists $n$ such that each of $P(n-1)$, $P(n)$, $P(n-1)$ is less than $n^u$. This was proved by Eggleton and Selfridge (Consecutive integers with no large prime factors, J. Austral. Math. Soc. Ser. A 22 (1976), 1–11). In fact their proof is constructive (see pp. 2-3 of their paper). It follows that then supremum in Question 1 is infinite, while the infimum in Question 2 is zero.
I should add that this phenomenon also holds for an arbitrary long string of consecutive integers. For example, there exists $n$ such that each of $P(n-50)$, $P(n-49)$, ..., $P(n+50)$ is less than $n^u$. This was proved by Balog and Wooley (On strings of consecutive integers with no large prime factors, J. Austral. Math. Soc. Ser. A 64 (1998), 266–276).