If we talk about a woman surrendering her virtue, it can mean that she has lost her virginitiy. However, it can also mean that she has committed adultery. The phrase is does not always apply to virginity. It is an old phrase - not used as much with regard to woman, but still hear occasionally. Another old phrase for a woman who was loose sexually was "a woman of easy virtue". Also, we can talk about people in general surrendering their virtue. In such a case it would mean the the person in question has done something counter to their standards or societies standards. I read a while back that some U.S. banks had surrendered their virtue, meaning that they had violated ethical banking standards. In this instance, it has nothing whatsover to do with virginity.

Yes it is a very old fashioned saying

It can mean a dozen things. Maybe for religious people it mean "losing your virginity". But it can also mean "selling out your principles".

No, I've never heard the phrase "surrender one's virtue" but it might mean just giving up like in battle or war. Lose one's virginity as you probably knows means no longer a 处女/男. Some similar phrases are "first time" and for a girl, "popping her cherry"