I had it happen to me 2 weeks ago, same symptoms.I took out the HDD and hooked in into my desktop and ran diagnostics. She was bad. My ram was bad as well, put in new HDD and ram, now its running good.
My initial thought was that the HDD is dead. But when I tried running the hard diagnostic from BIOS, the result is that the hard disk is in good condition (for both the quick and comprehensive testing).
This sounds like more than a drive issue, the laptop should not turn on soon as you connect the power supply, as the battery, if working, would be getting it power.
Can you run the laptop on only the battery? Looks to me like you are having serious motherboard issues with either the power connectors or possibly some damaged componenets. Your laptop looks to be about 4 years old, so all the moving around it does could have easily damaged it enough so it does what you describe.