Hey MrtBahgs,
You've got a bad RAM card. No doubt about it. If you replace just one card make sure the stats match for optimal performance. If you want to replace both you have a bit more room for upgrading. To be able to tell you what you can upgrade to I would need your exact product number. For information on finding your product and model numbers click here: How Do I Find My Model Number or Product Number?
If your notebook is a custom built (you chose the hardware), I will need you to send me your serial number in a private message so I can check which motherboard it has.
Your HDD seems fine if it doesn't detect corrupted info. I wouldn't worry about the chkdks /r, it does the same thing as /f, except /f allows you the option to reboot and lock the drive, where /r will fail if the drive isn't locked. Event Viewer is the correct location for viewing the chkdsk information. It can be found titled as chkdsk and wininit.
For information on using the HP Recovery Manager, you can check out this guide here: Performing an HP System Recovery (Windows XP). I would suggest replacing the RAM before resorting to this as it doesn't appear you have any issues with the actual operating system or HDD.
Regarding the battery. Can you tell me if the notebook runs with the battery installed? It is possible there is a power leak in one of the cells that can cause all sorts of wonky issues. Replacing the battery of a notebook this old is not a bad idea.
Hopefully I didn't miss any of your questions! If I did feel free to respond in all Caps. 