Translation:Η γυναίκα πηγαίνει έξω.
"Η γυναίκα βγαίνει έξω." --- Is this wrong?
I think they are different phonetic spellings: πηγ = βγ. We'll need to hear from a native speaker regarding which spelling is more common now.
Is there a difference between 'πηγαίνει' and 'πάει'?
No difference. You can use both.
i tried "η γυναίκα πάει έξω" but it was marked wrong