Translation:Will you marry me?
After this lesson I used this phrase to propose to my fiancee. We both study Mandarin on Duo. 她说是的!
I think the "set phrase" is a reference to wedding vows where 愿意 is the equivalent of English "I do."
In Chinese, until recently there was arranged marriages, then differences for men and women:
求婚（女性）：嫁给我，好吗？or 你愿意嫁给我吗？which conforms to the traditional "the man is the breadwinner and head of the household" mentality
（男性）：娶（了）我，好吗？or 你愿意娶我吗？ which sounds a bit strange, so the gender neutral 你愿意跟我结婚吗？is preferred.
Probably, but don't think that matters. Where's the grammatical error? Here's an example of it being used in one version of "Beauty and the Beast":
After that Beauty lived there in the Beast’s castle and was well content. Every day she went out into the gardens, and the Beast came and played with her for awhile, and she grew very fond of him. Every day before he left her he said, “Beauty, are you willing to marry me?”
But always Beauty answered, “No, dear Beast, I do not wish to marry you.”
Then the Beast would sigh heavily and go away.
...if you were reading that, would you really have a problem with it?
"Are you willing" and "Will you" are not mutually inclusive. One may marry whilst being unwilling to do so.
In this case, I think you would say "我愿意。" Since Chinese doesn't really have a word for "yes," it is common to answer in the affirmative by repeating the verb used in the original question.
In Taiwan this sentence will be "妳願意嫁給我嗎？"
好 means yes. In every yes or no question, you can reply 好 or 不好. But this one say 我願意 is better, I don't know why every excited person says 我願意 in this situation. But if you are not very excited when you're proposed, well you say 好 is definitely accepted. Different word shows different emotion.
i think "are you willing to marry me?" is a better translation than "will you marry me?"
Isn't the speaker studying a broad in Australia? XD