I'm not sure doing this with Safari is possible, but there is kiosk software out there that should fill your needs. They all seem to use WebKit, so the browsing experience will be similar to Safari.
Here's what I found:
- eCrisper $79. Looks fairly configurable, it would allow you to display some branding or supplementary information in addition to the browser.
- xStand $99.99. This looks like the most actively developed and supported option. Available on the Mac App Store.
- Kiosk $4.99 This is the cheapest ($4.99), but also has the fewest features, and hasn't been updated since October 2011. Available on the Mac App Store.
eCrisper and xStand both have demos available.
Any of those should fulfill requirements 2 and 3. For the first one, I'm not aware of any way to boot up a Mac automatically when it gets plugged in. There is an option in Energy Saver Preferences to "Start up automatically after a power failure", which will restart automatically if the power gets disconnected, but for a cold start, you need to press the power button or use Wake on LAN.
However once the power goes on, you can put whatever kiosk software you decide on in the login items, and enable automatic login (probably for a user with limited privileges, as an added security measure). That will get it automatically booting into the locked down browser mode.