IPhone 3Gs turns on and off itself when being charged

No, the iPhone does not self-condition. Apple has a guide for conditioning laptop batteries, yet I can't find a version for iOS devices (still looking).

There is a general tips page on maintaining great battery life, as well as general terms about battery replacement.

Personally: This sounds like a bad battery. It sounds like it just plainly is not holding a charge after a certain discharge point. Remember that iOS multitasking is not true multitasking. Barring streaming audio (actually persistent audio in general), there is (almost) NOTHING that your phone can constantly do for more than 20 minutes at a time.

I've streamed audio for hours (hours being... maybe 5?) during the day at ~40% and it did get under 20% by the time I was done, this was both on an iPad, and roughly equivalent on an iPhone 4. Connected WiFi, 3G (on the iPhone) enabled but not being the active radio in use, of course.

I don't remember ever reading about a problem with proximity sensors in iPhone 3GS running iOS 4.x. The well-known problem was with iPhone 4, and it was fixed in iOS 4.2.1.

I guess you will have to restore your iPhone, either to verify that your problem was caused by some rogue app you installed via jailbreak or to take the phone to an Apple Store.