The black light may not be the way to go. Frogs can see the light and it will make certain insects and objects glow. Could stress the frog. Ifrared is best for night time.

Your daylight bulb is most likely a UVA bulb. This particular bulb is most likely designed to simulate actual sunlight and is plant friendly. Most daylight bulbs are full spectrum though. Does it say anything about that?

Vitamins are fine, but you may want a calcium that already contains Vitamin D3. Fluker's makes a good calcium powder with D3.