"Neki van egy háza ebben a városban."

Translation: He has a house in this city.

[Kaja48]

What is wrong with she instead of he?

[jsiehler]

Nothing is wrong with it; Hungarian doesn't distinguish between "he" and "she". You can report it if it comes up again.

[Kaja48]

I will, thank you

[SedatKlc]

what is the conjugation for ön? neked?

[jsiehler]

The personal dative forms in the singular are nekem, neked, neki and for the polite "you", önnek or magának (which doesn't appear in this Duolingo course).

In general, ön and maga mostly just behave like ordinary, common nouns when you put endings on them.