OS install and Time Machine restore fail

I'm at my wit's end and hope someone here has any idea what's going on with my machine. I have a 2007 iMac (I know) that has been successfully running Snow Leopard for many years. Recently, I decided maybe it was time to upgrade and went to El Capitan. At first, things were fine, then it began started running super slow so I rebooted and it shut down during reboot. I called the support number and was guided through erasing the drive, restoring to Time Machine (with Snow Leopard) and reinstalling El Capitan. That worked great for about 3 days when I awoke to find a folder with a ? on my screen. Since then I have tried to boot into recovery mode and got a circle with a line through it (like a no smoking sign). Any attempt to reinstall Snow Leopard or restore from Time Machine failed. Disc Utility also fails with an "invalid node structure" error.

I gave up and made an appointment at the Genius bar. He tried to run diagnostics, but since the computer is so old, was only able to do one test. Everything came back with green check marks so he said it seemed the hard drive was ok. He erased the drive and tried to install the last 3 recent operating systems. All failed. He went all the way back to Snow Leopard and it worked! It seems the machine just wants to be in Snow Leopard and that's fine with me. I left with a working machine that just had to restore Time Machine files. I got home, hooked up Time Machine and started the migration. It got to 11 minutes left and stalled completely. Since then, I've tried erasing the disc several times and reinstalling Snow Leopard off the install disc. It fails with the error "installer could not copy necessary support files." Directly restoring from Time Machine fails at the 6.3% mark.

I'm banging my head against the wall here. Any help at all would be most greatly appreciated!