ipod classic won't mount

If you have not already, you can first try doing a Reset on the iPod

http://support.apple.com/kb/ht1320

You can try it while it is connected to one of the Macs. A Reset just "reboots" the iPod; it is not erased.

If that does not help, since it does not connect to iTunes on two separate computer, and it connected before, it's probably not a problem on the computer end. So that leaves the USB docking cable or a problem on the iPod. If you happen to have another docking cable, be sure to try it, or try the one you have with another iPod/iPhone, to rule out the docking cable as the problem.

If it's a problem on the iPod... I would try putting it into Disk Mode

http://support.apple.com/kb/ht1363

Connect the iPod, and try running both iTunes and Disk Utility to see if the iPod is recognized. If it is recognized, you may want to do a Restore on the iPod, because your problem may be cause by some type of data corruption on the iPod's hard drive. A Restore DOES erase the iPod's hard drive, then re-installs its software and sets it to default settings. That may also resolve the " inserting a short pause between tracks" problem.

It sounds like the iPod hard drive may have gone bad. In Disk Utility, you should see the iPod and its "disk" (volume) indented below, in the sidebar. If there was a very serious data corruption problem, it's possible for the disk to not appear there, but you should be able to select the iPod (the device) in the sidebar and reformat it on the Erase tab. Since it is "greyed-out," I think that means Disk Utility sees it as a device, but its actual storage is not accessible. The other problem you mentioned may have been an early sign of a hardware problem; the iPod may have been having difficulty reliably accessing its hard drive, which caused the pause between tracks.