I don't think you can ask iTunes *not* to sync,, it will automatically sync every now and then to make sure all music in the cloud is identical to playlist in iTunes.
Sure you can tell it to not sync: option for it under a device's Summary panel in iTunes. I have my iP* devices setup that way, and they only sync when I click a button. But yes, first time a device talks to a new computer, iTunes will try to sync.
As to why music is being erased, unsure: I have plugged my iPhone into numerous computers over the years and saying "Trust this Computer" "Do not allow" has kept my library intact.
One thing that is possible: something on the iPod is corrupt re: trust/no-trust settings for computers the iPod has plugged into. Go to Settings > General > Reset > Reset Network Settings to get the trust/not-trust settings back to square one.
If not that, MAYBE the version of iTunes on the computers you are trying to plug into are downlevel and bringing issues to the iPod? OP has not stated what version of iOS is on the iPod, assuming 11.X since that's the sub-board, but could be incompatibility issues between old iTunes and new iOS.
Given that the friends have Macs and apparently Apple cables, why not just use the cable and plug into whatever Apple charging bricks/cubes they must have received with their devices (and hence having the cables) that are lying around?