I would like to buy the device for my child: is it possible to lock the apps, including Safari, so that a child can use it purely as an iPod with Spotify, for example, but cannot access the internet separately and download apps themselves?
Can I limit the play time but leave the music listening time open? Thank you for your input....

Yes, most apps for the iPhone also run on the iPod.
If you are interested in a certain app, you can also select the app in the App Store and look under Compatibility.