Translation:The famous actress lives here in one of the houses.
Not much. "Az egyik" sounds a bit more definite. For example, if I knew which house it was, I would probably say "az egyik". If I didn't know exactly, I would probably say "valamelyik".
"Egyik" is also the one used for "one of the two". As a contrast to "the other one".
Also, "egyik" definitely refers to just one thing. "Valamelyik" may refer to more than one: some.
Az asztalon 10 pohár van. - There are 10 glasses on the table.
Az egyikben víz van. - One of them has water in it.
Valamelyikben víz van. - One/Some of them has/have water in it/them.
In earlier times it was common in English to say "some one of." "Some" meaning "an unspecified member or members of a group of things."
Could one say "...a valamelyik házban"? It feels like the "a" is missing here because we have "the" in the English sentence.
No, not really. The translation of valamelyik as "one of the" is a bit misleading. The "the" refers to a certain group of houses, but you don't know exactly which house in particular, so there's no a in the Hungarian translation.
Yes, "valamelyik" is definitely indefinite, which is why we can't have a definite article in front of it.
Some of these "vala-" words can also be used in a definite sense. Like when we are talking about "that (special) someone". "Az a valaki" needs the definite article because of that certain rule about demonstratives.
I think this sentence can be more literally translated as The famous actress lives here in some house, which is maybe not as natural in English, but explains why there is no the.
To explain further, Hungarian does not really translate some directly into vala-; it needs a suffix, like valami or valamelyik. How do we know which suffix to use?
Remember that English also makes this distinction sometimes: we could say What house does he live in?, but it is better to say Which house does he live in?, because which house? means which one of the houses?.
So which is the right pronoun to use with house. Same in Hungarian: melyik ház for which house, and so some house is valamelyik ház (with the meaning one of the houses)
If there were such a word as "somewhich", I think that would be a good translation of "valamelyik". It is an unspecified one of the houses.