Can I switch a pulse solenoid from the onboard 5V/3.3V power source?
1) Can I send a 3.6V solenoid a 60ms pulse to open/shut it from a 5V
  power source without burning out the coil or causing any other damage?


No. If it's 3.6V then you need 3.6V - 5V will most likely make it smoke. Use a regulator (preferably a buck one, since it is apparently 500mA) to power your H-bridge.


  2) Is 3.3V too low to power a 3.6V solenoid?


Maybe, maybe not. It may be that 3.6V is the absolute maximum and may switch from 3V to 3.6V.


  3) Do pulse solenoids still generate back EMF?


Normally, yes, since they are just a coil like any other inductive load. Your H-bridge should deal with that though if designed right.

To get any meaningful answers you will need a datasheet. Buying such things from eBay is never a good idea unless you have used it before and know what it is, what its specifications are, and how you use it, or they provide a link to a datasheet. Without a datasheet most questions will be answered with a "maybe".