Recall that a door space is a topological space where every set is either open or closed (or both). A topological space is finite if it has finitely many points. I'm interested in learning about finite door spaces.

Question 0: What is an example of a finite topological space which is $T_0$ but not a door space?

Question 1: Let $n$ be a natural number. How many door topologies are there on a set with $n$ elements?

I'm interested in both the "labeled" and "unlabeled" versions of question 1.

Recall that via the specialization order, finite topological spaces are equivalent to finite posets.

Question 2: Which finite posets are the specialization order of a door space?