"The lawyer reads the private answer."
Translation:Ο δικηγόρος διαβάζει την ιδιωτική απάντηση.
I thought δικηγόρος was one of those words that does not change for masculine or feminine (ο δικηγόρος, η δικηγόρος) so I am not sure why the feminine option was marked incorrect in the multiple choice. In other words," Ο δικηγόρος διαβάζει την ιδιωτική απάντηση" was accepted but "H δικηγόρος διαβάζει την ιδιωτική απάντηση" was not.
Is there any rule about when I should use ᾽ι᾽ and when I should use ᾽η᾽ or indeed ᾽υ᾽
It's like the rule in English about when to use "ee" (e.g. "green") and when to use "ea" (e.g. "clean").
If you know the pronunciation of Ancient Greek or Old English, it's easy to remember -- since those different spellings correspond to what used to be different pronunciations.
Now that the pronunciations have merged, I'm afraid you're going to have to memorise the spelling. There's no rule.
Do "διαβαζει" always push the next part of the sentence into akkusativ?
You can use the verb without a direct object (as in English): Διαβάζει. "He is reading."
When the direct object is a pronoun, it comes before the verb: Τη διαβάζει. "He is reading it." (e.g. την εφημερίδα "the newspaper", which is feminine).
If a direct object is present, it's in the accusative case, but the direct object is not necessarily "the next part of the sentence", i.e. after the verb.