Typically your Windows product key can be found on a sticker on the installation CD case (which of course you don't have) or on a certificate of authenticity card. You can also find the product key on a small Microsoft sticker on the outside of your computers case.
For the moment the only thing you need to "save your stuff" is a blank cdr or dvd and a cdr/dvd burning program. Just burn all you want to save to a disk.
Yes, to repair you will or do need a Windows Disk. HP should be able to explain to you either how to obtain one OR how to copy one from the other partition that PP spoke about in his post.
Don't give up, CALL or Chat with HP. I am certain they will be able to help you. I think there is probably a good chance you can repair this and not have to totally reformat. But wait until you get the advice from HP. Go with them FIRST. They will give you the best options AND will be able to keep your warranty valid. Don't do anything drastic which could void the warranty. You really should try that chatline first, they may be able to walk you through what you need to do right away. They are good, at least I have had very good luck with them. If they feel you need to actually speak with a tech they will give you any additional number you will need.
Don't forget, we are here if you need us.
Judy