no

DataLog is used to verify smart contracts. I don't know any specific examples of prolog used in the industry, though I am sure there are a few. Probably, there are expert systems implemented with prolog that are still used.

Logic programming in general has a few usages, the unification algorithm is used for pretty much every type system. Also, it is quite good for verification systems. For example, I know some symbolic execution systems implemented in OCaml.

mother can be used in several ways. If both X and Y variables are uninitialized, then it looks for all mother relationships. If one of them is initialized, it looks for matching relationships. If both are initialized, it returns true if such a relationship exists.

Yeah, the Y is a wildcard in that position. Typically, you would write it as an underscore, primarily because most Prolog compilers will warn about unknown variables, since those could also just be a typo of an existing variable.

Depends on how you want to define your domain knowledge.

The thing you need to define for sure is the predicate mother/2 (Which has arity 2, or in other words, two arguments). From then on, multiple options are available:

1. Take mother(X, Y) as an "axiom", and define mother terms for all elements:

mother(abel, eve).
mother(isaac, sarah).

2. Derive mother(X, Y) from female(X) and parent(X, Y) terms.

mother(X, Y) :- 
  parent(X, Y), 
  female(Y).

3. Smash the institutional gender power structures and define only parent/2 terms instead of mother/2 and father/2.

doesn’t this definition lack a recursive case to ensure that the mother is either Eve or a descendent of Eve

We don’t see the definition of mother. It might already encode that Y is a person.

And there should probably be a father case in there as well?

While every person does also have a father, it’s completely redundant, since being a person can fully be described by [Edit: ~~being~~ having] a mother (or being Adam or Eve).