"They read me a book."
Translation:Αυτοί μου διαβάζουν ένα βιβλίο.
It shouldn't - it's a mistake.
It's fixed in the next version of the course that is currently under development.
Why μου is used here? It is weak genitive of εγώ (according to wiktionary, BTW, what are weak and strong cases for?) As far as I know, languages that have dative case use it in such sentences. But Greek doesn't have it.
Yes, Greek lost the dative case except in a few fixed expressions.
In standard Greek, it was replaced by the genitive case when using pronouns, but by the accusative case in the north. (When using nouns, then one usually uses the preposition σε + accusative.)
So you will find things such as μου δίνει, μου λέει, μου διαβάζει for "he gives me, he tells me, he reads to me" with genitive in standard Greek for what would be dative in other languages.
I'm not sure whether weak/strong is standard terminology, but the short forms of pronouns are used in close connection with a verb: right before it in the indicative, right after it in the imperative.
The long forms of pronouns are used by themselves (e.g. as the answer to a question), as the object of a preposition, or for emphasis (e.g. "he sees ME [and not anyone else]").
French has similar double forms for its pronouns.
I indicated the used pronoun: