Well, I hate to say this as I love diagnosing problems and feel accomplished resolving them, but the laptop may be beyond basic solutions at this point. I had a Gateway laptop that was doing similar problems and I was completely unable to resolve it. I concluded it was motherboard failure.
However I have come across a few HP laptops that had similar symptoms and upon replacing the battery the computer booted up like nothing was wrong. If you don't mind throwing a little money at it you could always find a cheap used or replacement battery from eBay to see if that would resolve the issue. If I was present I would eventually take the laptop apart and start with the montherboard, plugging up the basic components needed to run; screen, processor, and one stick of RAM..
If you arent tech savvy enough to do that I may take it by a computer store to have them diagnose it, some will do this for free and not charge you if they cant figure it out. Sometimes its better to have a tech present to diagnose stuff as they can physically see whats going on.Only thing at this point would be to try one stick of RAM , then test with another slot. Wish I could be of more help..