If a matching file already exists in iTunes, it will not be copied from the device. Instead, the existing file in iTunes will be used to re-create an entry in a playlist.
So you can just select the playlist(s) you want to re-create and click the Transfer-button, done.
No. Using the Sync- and Transfer features nothing will be deleted. Not from the device, and not from iTunes.
Unlike iTunes' strict mirror-sync, copyThing does an add-sync, adding missing items only and leaving other content intact.
Yes. In the Tools window, you can delete "dead tracks" - media items with broken links to the actual files on disk.
You can also remove playlists with a given criteria (this will not remove any media items in iTunes - only playlists and playlist entries).
If multiple devices are connected, just choose the desired one from the popup menu in the upper-left area.
Disconnect your device from the USB cable, unlock your device by entering your passcode on the lock screen, then re-connect the USB cable.
If you're using a cheap USB cable for your iPhone/iPod/iPad, that may very well be the reason. Try the original USB cable from Apple.