Do I need to use the QR Skewer that came with my trainer? What sort of lubricant do I need to use?
You should use the QR skewer which came with the trainer for several reasons:

1) Its cheap and ruining it (rather than your own skewer) will keep your nice skewer which came with the bike in good condition. It may also be stronger than your (possibly boutique) skewer in the trainer. (Safety) 

2) It has markings which can help bite into the cups of the trainer which attach to the skewer. (Safety)

3) Liability -- the manufacturer said to put it there. If you don't, you're SOL if it goes wrong with the skewer. Plus, it takes about 5 seconds to swap the skewers so why not do it?

That being said, as you found out, many enclosed cam QR skewers will fit into trainers, but why risk it? I would not recommend it (as the trainer manufacturer probably wouldn't either). 

The lubrication of the QR skewer is for the cam - you don't need to do anything for a new skewer.