"Also sometimes the plane turns out itself" Do you mean the plane turns its engines or power off or both? The FPS with the 737 would depend on the definition of the livery. As long as the FPS stay within acceptable range it should not matter. If they don't then just use lower definition liveries.
Yes that would be nice if Laminar could fully save a situation. Whenever I have tried it the aircraft is back at cold and dark. The way I get round this is to pause the sim then hibernate the computer. If your computer can hibernate/resume without any problems, use that.