I'm on 3.1.2 right now so my questions are, if I do Erase All Content and Settings, would that be the same as a restore?
No. Erase all content and settings is exactly as provided - it erases all content and settings only, which is a secure erase.
Would it keep me on 3.1.2 afterwards, and would it require me to plug the phone into a computer afterwards?
The firmware/OS is not touched. This erases all content and settings only. Your iPhone will be completely erased of everything but the firmware/OS.
If something goes wrong with this process for whatever reason, you will need to have a computer with iTunes available in order to restore the iPhone with iTunes if needed.