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
This answer is already accepted.
what's the difference between βγαίνει and πηγαίνει?
"βγαίνω" means "I am going out". ("Do you go out often?", is "Βγαίνεις συχνά;")
"πηγαίνω/πάω" is "I am going"
"έρχομαι" is "I am coming"
"φεύγω" is "I am leaving"
According to Google Translate the difference is "come" versus "go". Also I have seen at least one internet page that uses the words that way, with βγαίνει for come and πηγαίνει for go, in a single sentence.
"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.
That's correct as well.