Is it common to see some eye floaters when looking at the sky?

Yes, sometimes. Floaters are due to imperfections in the clear gel inside the eyeball, and tend to become noticeable as a person ages. Floaters occur in many people, and sometimes are more noticeable if one is looking at a plain blank surface such as a blue sky. Large numbers of floaters, noticeably increasing floaters, flashes of light, changes in vision, etc..., should be evaluated by an md eye doctor.
Yes. They are either tear film components or blood cells of retina being seen.