1. Yes.
2. No.
Photons are massless. However, light still does have momentum. p = mv applies to objects not moving at speeds approaching c. You might recall Einstein's E = mc^2. As I understand it, this is a shortened form of the full equation:
E^2 = (pc)^2 + (mc^2)^2
In which the ability for light to have momentum is seen even though it has no mass.