Long answer: The subject is usually pronounced with a different intonation than the verb. The subject raises in tone and then drops again when the verb starts (and raises again). The "της" in "ο άνδρας της διαβάζει" therefore is pronounced in the same tone as "-ρας", while in "η άνδρας τής διαβάζει" it has the same tone as "δια-", so a lower tone. But all of this doesn't differ much from english or most european languages, so it should just come intuitively, so:
Short answer: don't worry.
When you have two possible words that are correct individually but not together that is considered a mistake. Here is an example:
The word "He" is correct right?
"eat" is also correct again right?
But put them together and say "He eat" Is that correct? No.
Well in Greek
"η" is a word ("the" for feminine nouns)
"άνδρας" ("man" masculine noun) is correct
But put them together and they are wrong. You can't use η with a masculine noun.
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!
No, There two kinds of της so to speak, the possesive and the inderect object. One needs an accent mark (τής) and the other doesn't. When you use it to denote ownership (possesion) it goes unmarked. So "ο αντρας της διαβαζει" means "her man reads". Now, if you use it as an inderect object, it needs the accent mark, so "ο αντρας τής διαβαζει" means "the man reads to her".
You can always use both. In case you want to say "her man reads to her" it would be "ο αντρας της τής διαβαζει".
Hope I didn't bewildered you with my messy explenation. xD
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.