Most laptops that I've seen will shut off if you close the lid all the way. I don't know if that's a universal thing, or just the few I've looked at.
Leaving the lid up a bit also helps keep you from tossing something on top - which could cause it to heat up.
It's not really too likely that OE itself is causing problems. It's a pretty stable part of standard Windows software. OE may trigger a problem when you try to open it, because of your firewall, or because the connection it tries to make isn't set up right.
If you can get OE open, try going to Tools|Options and click the "General" tab. At the bottom there should be a "This program is the default email program" and a separate "This program is not the default email program." Click the "Make Default" button next to the "is not the default" if you want to make sure it won't try to connect itself up according to some schedule or other such setting you've put in the setup.
(Yes - make the "default" be "not be the default program" sounds a bit strange, but that's how it works.)