Copy only folders over?
 

I don't know what you're planning here, but what are "OS X files"? In other words, you have Applications. Are those OS X files? If you don't copy /Library, they won't work. But there are OS X files there. What about /System/Library? Various configuration directories?

What are you trying to achieve?

Since the boot loader and the kernel do not match, I was going to reinstall OS X onto a formatted backup drive so that it at least boots normally again.

Then I was going to copy over everything else (all non-OS X files) onto it in hopes that it would bring me back to a working backup drive again....

Why not reinstall over what you have?

The last time I tried that it went back to not booting again. So I'm thinking "something" (core services) is bad on the main drive and when I copy it over it keeps the backup drive from booting also.

But that entire folder would be replaced during an OS reinstall.