If I had no interest in the HP tools or recovery partitions, I'd get rid of them.
As far as I know, there is nothing in either of them that is needed for Windows to boot or run properly, but you might get a second opinion.
Your "system" partition is marked as both active and system.
You say you have already made recovery disks and that your PC is out of warranty.
I'd also download a Windows 7 ISO and burn it to a disk so I could do a clean install using the existing Product Key.