iPod 5th Gen (A1136) unmounts itself on Macbook and iMac when not syncing.

Have you enabled the drive as a storage device?

Using your iPod as a storage drive

If the iPod unmounts to quickly, you need to manually force the iPod into Disk Mode using the instructions in this Apple support document.

Putting iPod into Disk Mode

B-rock

That is "by design" so that the user can disconnect the iPod without ejecting it first. To keep the iPod mounted whenever it is connected, you need to check the checkbox on the iPod's Summary tab, for Enable Disk Use. But when you do this, be sure to remember to eject the iPod (using the small button next to the iPod's name in the iTunes sidebar), before physically disconnecting it.

FYI for this

I want to load Mac OS X onto the iPod and use it as a boot loader in order to install Mac OS X on another machine. It's the only option I have since I do not own a flash drive, or external drive.

If you want to do this with installing Lion or Mountain Lion, there's a far easier way, if the "target" Mac already has at least Snow Leopard (or Lion) currently installed. Just run the Mac App Store on it, and log in with the same Apple ID used to purchase the Lion or Mountain Lion upgrade. You should be able to (re)download the Mac OS X installer on that Mac, like you can with any previously purchased app (on the Purchases tab) Then, just run the Installer on that Mac.

Glad to hear it was resolved. 🙂

B-rock