I have woken up with a black dot on my eye?

Black eye dot. If you have a new pigmented lesion on your eye, you should be evaluated by an eye doctor.
Hard to say. Likely something benign like a burst capillary. However, if there's any change in vision or pain get it checked out right away. When dealing with eyes i tend to lean towards the more cautious approach. Eyes are precious and i don't mess around with them.