First, yes your yeast is fine. (assuming you don't keep your house at 120F)
The Propagator pack is meant to be used with a starter. So in this case he severely underpitched, so expect it to take 72+ hrs to show any signs of activity. Checking the gravity with a hydrometer is the only way to determine if there is activity. The airlock is there to let CO2 release and it only shows bubbles if there is a great seal on the bung/lid.
The Activator pack has 4x as many yeast cells as the propagator and it is still recommended to always make a starter with liquid yeast to ensure proper pitching rates and to prove yeast viability.