My child received an iPod touch as a gift but it contains the giver's Apple ID, email, etc. They put music on it for my child under that ID. We want to bring it into the family account for monitoring and such. Can I reset the device and assign a new Apple ID for the child? And should I assume any purchases made under the old Apple ID are lost forever? (We have no access to the account.)
To start, let's talk about the music on it. The music is sort of tied to the Apple ID (not iPod itself). By "sort of", I mean you need that Apple ID to download it, but signing out of the Apple ID and into a new one won't delete it. For example, if you were to purchase music under one Apple ID and then signed out of it and into another, the music would remain on iPod, but if the music was then deleted (like when iPod is reset), you would be unable to re-download it, since it's not tied to iPod itself.
What you need to do is use iTunes to sync iPod to the iTunes Library on your computer. Once you're sure the songs (and anything else you want to keep) are all there, you can go ahead and reset it.
If it's running iOS 6.1.6 or earlier (to check, go to Settings > General > About, then look under Version), then yes, you can reset it and assign a new Apple ID to it without any problems. If it's running iOS 7.0 or later, however, it depends on whether or not Find My iPod is enabled (to check, Settings > iCloud, and look for Find My iPod at the bottom). If so, resetting it will trigger the device's Activation Lock and you won't be able to set it up. In that case, you will need the Apple ID password of whomever gave your child this iPod to disable Find My iPod, but once it's off, you can go ahead and reset it.
Once it's reset, use iTunes to transfer whatever content you want back to it.
If you have any questions about any of this, just comment them below my answer.