Hi Zack, just came across this thread today, very cool project. I'm in Bowie, just bought a Premium GT a few weeks ago. If you need to take any further measurements or whatnot just shoot me an email and I'll send you my number and you could swing by sometime.
John, this is really strange because I have exactly the same two issues with my GT. Starts great cold but drive it for 10-15 minutes and shut it off and restart and it sounds like its not going to kick over sometimes. The strange thing is that my remote start doesn't work either on both keys...