Ion drives do exist. They simply ionize a gas plasma which propels the ship forward as the charged particles are accelerated away from the ship. I think they have gotten a whopping 2.5 lbs. of thrust from them (but it is very efficient).
As I understand the idea of folding space, you can actually arrive at your destination before your light beam, there is no reason you couldn't, you have not surpassed the speed of light, you are just appearing at your destination.
The real conundrum comes when you enter the fold at near relativistic speeds, and the destination point is in your approach path. It is possible to time it just right so that you hit yourself, thus preventing your entry in the first place. If that doesn't make your head hurt, I don't know what will...
