Some Mormon girls succumb to worldly laziness. Her church is more important to her than you are ever will beplain and simple. If she is motivated enough to want to go on a mission, she will likely not settle for having a second class "eternal companion" you who is a convert or one who cannot be sealed to her for eternity. Take issues as they arise, rather than throwing away what could be a good experience for you and her based off of what might happen. Put your best foot forward; be soft-spoken, courteous, well-mannered, chivalrous, and respectful. Mormons are very strict about avoiding sex outside of marriage.
While you will be of the right age to date, the Church discourages you from trying to date someone while on your mission trip. They don't have any time. I remember one time I did just that with a group of friends and someone saying, "Oh you have it so rough. She said we couldn't watch it because the lady who produced the doc was formerly a professor at BYU but has since left the church, so she wouldn't have anything good to say. I have been a doctor's wife for 1 year now and we've been together for 7 years. Disney gives them the fantasy where they can easily achieve a standard of perfection with low risk. I then asked some questions about some of the essays, and her answer to everything is "because god.
I told him that this wasn't working for me. I thought I was very understanding and, while I did invite her out at other times, I was very clear that her studies were paramount and I wanted to support her as best as I could during this time. I would love to someday find myself sitting in the pew with you, sharing this amazing journey.
It certainly isn't easy. Found this blog just now after searching "married to a surgeon". There is no question that God loves all of His children, and that obviously includes non-members. Is it naive to think we could raise our children to fully participate in two different faiths. A lot of people will tell you to run but if she is in her late 20s most Mormon guys her age are married. Like it was mentioned above, see how she deals some of the issues now and if she can't handle it, it might be better to find someone else.