how can the answer be both, 'her husband reads a newspaper' and 'her husband reads her a newspaper'?
It can't really, this should only be "her husband reads a newspaper".
Her husband reads her a newspaper = Ο άντρας της της διαβάζει μια εφημερίδα.
The man reads her a newspaper.= Ο άντρας τής διαβάζει μια εφημερίδα.
Please see my comment here for another example and the explanation on how words are grouped together just by adding an accent!
It can be either "her husband reads a newspaper" or "the man reads her a newspaper". This ambiguity emerges from the fact that in Greek ο άντρας means just "the man", but also can mean "husband" in context.
This sentence can't be translated as "The husband reads her a newspaper". I don't know why they have it like that :/
Ι wrote " τις"; it could be : the man reads them a newspaper ? because I heard: " tisse"
The man reads them a newspaper=Ο άντρας τούς διαβάζει μία εφημερίδα.
what's the difference between τους τις και τα ? can you make sentences with those words please
The answer to this doesn't come from a single grammar lesson. It's a combination of the definite article, the personal pronoun and the possessives. Read this about the definite articles. As for the rest, please check out the "possessives" and the "personal pronouns" skill. There are many available discussions you can access by means of the search bar in the Greek discussions main menu.