How can I listen to just part of a single song, or just skip some time?
If you have access to a Mac, you can right-click the song, then go into Get Info → Options, where you can then set both Start and Stop times.

These preferences will propagate to your iOS devices next time you synchronize them to your iTunes Music Library. If you subscribe to iTunes in the Cloud (via iTunes Match or MUSIC), that will be done over the air.

This is a persistent modification that will apply every time you listen to that song (until you reset the Start and Stop times).