Number of phone locking patterns

There have been many attempts at computing this number. Most famously this video which appears to show every possible combination.

All of these report the number as: 389,112

A team of researchers from the University of Pennsylvania looking into how these types of passwords could be discerned through fingerprint smudges also calculated the same number (turns out they were also the first to research these kind of "Smudge Attacks").

I don't believe there is a "nice" (non-exhaustive) method of finding this number. The paper on "Smudge Attacks" also states:

Due to the complexity of the intermediate contact point restriction, we calculated this result via brute force methods.

While this is not a proof, I'd like to provide some intuition as to why this would only have solutions based on exhaustive searches.

A smaller version of this problem would be counting the number of Hamiltonian Paths (a path that visits every dot exactly once) of the following graph (which ignores the "Once a dot is lit, you can use it to reach another unlit dot" condition):

However, counting Hamiltonian path for most graphs usually requires an exhaustive search.

the problem of finding a Hamiltonian path is NP-complete, so the only known way to determine whether a given general graph has a Hamiltonian path is to undertake an exhaustive search (Wolfram Mathworld)

So counting all passwords which use all 9 dots probably requires an exhaustive search.

So to answer your first question: No it's unlikely there is a nice closed-form calculation to the problem that isn't exhaustive.

As to your second question: I believe psychological factors may be more significant. A paper from 2014, showed a lot of passwords only used 4 dots, and many passwords were just alphabetic letters like C, N, O, etc.

If you'd like to know if there are more "rigid structure" passwords than "non-rigid structure" passwords, you'd need to define "rigid structure" more precisely.