1. You can have non-zero voltage and zero current if there is insulating material (e.g. air) between the two points. This is the case, for example, on a Van de Graaff generator which has a high voltage due to static electricity (non moving electricity = no current).
2. If you have no voltage across a circuit element, then you don't have anything to drive a current. Conventionally speaking, from an electrical engineering standpoint, the answer is probably no. However, you can have charges (say free-floating charges) freely moving in a direction which would then be called a current. For example, in a cathode ray tube, you accelerate the charges over a short distance (using a voltage) and then they basically free-stream to the other side of the tube. During it's free-stream period, there's no voltage difference, but there's a current.
Voltage is the potential energy (per unit charge) stored between a configuration of charges. This potential energy is what can accelerate charges from one place to another.
Current is just the movement of charges (how much charge passes a point in space per unit time).