Can I simply install a fresh OS X without a thumb drive? I've always installed a new OS X install via a thumb drive. However, at that time, I only had a single hard drive. Now, I have two hard drives, but the question is: Do I still need a thumb drive now?

I would say I simply copy my main installation to my second hard drive (Carbon Copy Cloner?) and then format the main drive and simply run the installer, stored on the second drive and install OS X on the first drive. No need for a thumb drive, right?
That should be possible. At least it was possible for previous OS X's, but I don't see why they changed it. So yes, you can run the installer from the second hard drive onto the first one. You can even install any other OS X onto another partition on the same hard drive.