I have a moving black spot in my right eye. It's small but annoying when I move my eye around. Will it go away or do I need to see a doc? I am 22.

Need to see MD. It sounds like you have a new-onset floater. This is part of a natural occurrence called posterior vitreous detachment, where the gel inside the eye separates off from the retina, sometimes forming visible clumps in the process. Because this process can sometimes cause tears in the peripheral retina which can lead to retinal detachment (a vision-threatening thing), you should be seen asap.
Depends. This sounds like a floater. If this is something that has been present for a while, then it does not need evaluation. A new floater does need evaluation. See an ophthalmologist.