I was marked wrong for 'I feel pain' which is a customary British way for expressing the idea (along with 'I feel pain). We don't really say 'I hurt' unless one is a young child.
This really ought to be changed. In Britain we would never say 'I hurt'. It sounds very odd to us. 'I feel pain' is much more idiomatic and should at the very least be an alternative.
Not at all, we have: πονάω/πονώ, πονάς, πονάει/πονά, πονάμε, πονάτε, πονάνε/πονούν these are the usual forms. Do not forget the accent!
How important are the accents? I understand for grammatical accuracy but if you had forgotten a few is it that vital?
Accents are a grammatical phenomenon and therefore they are vital. In some cases a change of acceny might change the meaning of a word. As long as you are learning the language it's worth it to make the extra effort to do it right.
Is "I am in pain" accepted? It sounds more appropriate for physical pain.
I don't know if you deleted this but I just wanted to say that: "I'm in pain." is an accepted translation here.