You can't format the drive or back up data if it won't let you boot to it. The only thing you can try doing is putting it in another machine and see if it will let you access it that way. If your not computer savvy, I suggest you take it in to have someone look at it for you.
sorry but no
The hard drive would still work if it was full. It sounds like you aren't computer savvy to follow my advice. It doesn't matter how old the drive is, it can go out the day you get it or whenever, i've already told you that. I've been troubleshooting computers for almost 15 years now. Take the drive in to a computer shop.
Looks like Hp wasn't very thorough. And how do you know that its not the motherboard thats bad? The next step would be to reset the cmos.