How does one test the freewheel? What are the failure modes (symptoms) of a failing or failed freewheel?

My understanding of normal free wheeling means the rear tire should rotate when the bike is moving and the pedals are or are not rotating. 

Failure would mean the bike and the pedals always move together, like a fixed or locked hub, or the pedals and cassette move but the rear tire seems disconnected.  

I intermittenly have a 1/4 - 1/2 crank turn 'slip' in the pedals before the wheel begins to move. The condition is very obvious on my trainer, not noticeable on the road. Is it a free wheel failure or something else? 

If the freewheel is indeed the problem as I suspect, is it repairable or am I required to replace the freewheel and cassette together.
You are correct about the two failure modes.  Try lubricating your freewheel with a light oil.  If it is dry or dirty that can cause it to not engage.  If that does not fix it then you will need to replace it.