Does the mineral calcite have a crystal lattice structure?
Yes. All minerals, by definition, have a distinct crystal structure. And yes, this is what causes the double refraction. Because of the way that the atoms are arranged within a calcite crystal, a light ray that enters the crystal from one side is slowed and “bent” as it passes through, and there end up being two rays that exit the crystal on the other side. One of these is slower than the other, and this is what causes you to see two “projections” of whatever you look at through the crystal. The first link below does a better job of explaining this property and how it relates to crystals, and the second provides more information about calcite in general. Hope that helps.