This seems to have been going on for a long time.
I know you are having daily Kernel Panics trebbler.
I would back up & erase the HD. Then reinstall the OS & see if it is any different with only default software. It could leave the Mac in an unbootable state if it KP's during reinstall or the erase etc. If you are lucky Target disk mode may work, allowing you to restore the (almost) working OS if you have a second Mac & correct cables.
How to use and troubleshoot FireWire target disk mode - Apple Support
You can try installing to an external disk if you really want. I would remove the internal HD if you are able to do that - just in case that is the reason for the KP's, if you have a USB/ Firewire caddy you can try booting from the internal HD when it is external.
I would also pull one RAM stick & retest out of desperation & then switch to the other one if it KP's - check how it needs to be arranged in the slots.
If it isn't any better it is time to bite the bullet & take it to an Apple store. They should try diagnostics - if they can't get the tools running something is just plain wrong with this Mac. They will charge for repairs which may be beyond the value of this Mac.
Sorry I have exhausted the steps I use to make AHT work.