April,
I'm glad to see about your interest in the program. For what it's worth, all of my applications were round 2 as I recall with maybe the exception of a program or two with commit dates after the LGO decision. LGO was my number one choice, so I based my application schedule around it.
As far as the engineering background, take a look at this link:
https://lgo.mit.edu/apply/qualifications/More directly, two years working experience and an undergraduate or graduate degree in engineering, computer science, biology, chemistry, or physics are required. Of course, exceptions are possible. This year we have an applied math and an architecture undergrad degree both represented. I hope this helps. Good luck with your applications!
-Robbie
LGO Fellow, Class of 2014