iPod classic 160 GB will not appear in iTunes

Does the iPod still work properly as an iPod when disconnected (play songs)?

Then, I even completely deleted iTunes and reinstalled it to my Mac.

Since you use a Mac and iTunes is part of the standard installation, how did your "completely delete iTunes"? What version of OS X do you run on the Mac?

Then I tried putting the iPod in disk mode.

What specifically happens when you run iTunes on the Mac and then connect the iPod in Disk Mode? What does the iPod's screen say when you connect it?

To delete iTunes I went into system preferences found where iTunes was, clicked the button that looks like a lock to unlock it, I then had to enter my username and password for the computer

I don't understand what you are describing about System Preferences. Which pane in System Preferences did you use?

then I exited out of that and dragged the iTunes icon from my desktop to trash.

From the Desktop? Why is iTunes on your Desktop? The iTunes app is located in the Applications folder.

NOTE: On a Mac, you do not need to delete the existing iTunes app. If you have a reason to install iTunes again, go here to download the latest version. Then run the installer.

http://www.apple.com/itunes/

For your actual problem...

When I connect it in disk mode it says "do not disconnect".

That's good. It means there is a data connection between the iPod and Mac. Was iTunes running when you connected the iPod? Normally, when you connect an iPod in Disk Mode (with iTunes running), iTunes should prompt you to do a Restore on the iPod.

It's still not clear to me how you "completely deleted iTunes and reinstalled it"... As the last step (for however you did it), did you download the latest version of iTunes here, then run the downloaded iTunes installer?

https://www.apple.com/itunes/

If iTunes is not properly installed, the background processes that identify when an iPod is connected may not be running.

Could you provide a link to those instructions?

Do you have access to another Mac? It would be helpful to narrow down the cause... Is the problem on the Mac side of the connection or on the iPod? If the iPod appears in iTunes on the other Mac, then the cause is more likely to be on your Mac. If the iPod does not appear in iTunes on the other Mac either, then the cause is more likely to be on your iPod.

Also, to rule out a faulty USB docking cable, try another cable (if you have one). Or try your current cable with another Apple device, to confirm it works.

You said the iPod still appeared on the Mac (although not in iTunes). With iTunes not running, run Disk Utility. Does the iPod appear in the Disk Utility sidebar? If it does, you can try reformatting (erasing) the iPod's hard drive using Disk Utility. This should be done in a particular way. What version of OS X do you run on your Mac? I ask, because the Disk Utility on El Capitan (latest OS X) was re-designed, so it works differently from previous versions.

If the iPod does appear in Disk Utility's sidebar...

NOTE: If you Erase the iPod's hard drive using Disk Utility, this obviously removes all of your songs (and other media) currently on the iPod. After the Erase in Disk Utility, you'll need to connect it to iTunes and do a Restore on the iPod, to re-install its onboard software. Then, you can re-sync your content back. However, you said your iPod currently operates as an iPod (plays its songs). If the iPod has an odd hardware problem that is preventing iTunes from recognizing it, you will not be able to do the Restore and re-sync your content.

To proceed, quit iTunes if it's running. Run Disk Utility. Select the iPod device in the Disk Utility sidebar. There may be a volume shown indented below the iPod device. Select the iPod device, NOT the volume indented below the device. Then, click Erase (along top of window). In the Erase setup pane, set it up like this

Name: [does not matter]

Format: OS X Extended (Journaled)

Scheme: Apple Partition Map

Click Erase. Disk Utility erases and reformats the iPod's hard drive. This should take less than one minute. If the process hangs or errors out, there may be a hardware problem.

After the Erase completes, quit Disk Utility. Run iTunes. Hopefully, the iPod device button appears on the horizontal bar. iTunes may not prompt you to do a Restore, but you need to do the Restore

Use iTunes on your Mac or PC to restore your iPhone, iPad, or iPod to factory settings - Apple Support

After the Restore, you can set up the iPod to sync like before.

That's a message from the OS X system (not directly related to iTunes) indicating there is data corruption on the iPod's drive. If your computer's drive had data corruption, you'd see a similar message, so it's not specific to iPods. (Your iPod is basically a mini hard drive with a built-in music player.)

Normally, when you put an iPod into Disk Mode and run iTunes, iTunes picks it up as an iPod in "recovery mode," and prompts you to d a Restore. Doing a Restore erases the iPod's drive, and that gets rid of the data corruption (if there's no hardware problem). For some reason, iTune is not recognizing your iPod. And that's why you may want to try doing the Erase using Disk Utility.

But, if there's a hardware problem that caused the data corruption, the Erase may fail with Disk Utility, or you may not be able to do the Restore and re-sync with iTunes.