Can iTunes pause a specified amount of time after a particular track?
You're asking two questions here. 


  It's possible to get iTunes to start a song in the middle?


Yes, for each individual track, you can press CMD+I (get info) and get to this tab:



Where you can check Start Time, Stop Time or both and specify a time. 
Part of a Gapless album is useful too, since it will not leave a gap time between tracks (nice when you have live albums or such).-

*Is it possible to get iTunes to pause for several seconds after playing a song?*


This is not natively possible within iTunes and you'll have to perform further research, try to find an App that does it and/or create some sort of Applescript. But doesn't seem like something natural to do, so it may not be easy to achieve (automatically).

A simple trick you could employ is to record a 30 second silence (you can probably find them if you google, it's used for ringtones) and play that in between your playlist.