Is it normal to be able to see the blood vessels in the eye?

In or on the eye? If you mean those that are on the white of the eye, they can certainly be prominent. If they have become this way recently, you should have your eyes examined. If you can see a branching pattern when your eye is rubbed (don't try this if you don't see it), you are experiencing the vessels of the retina in a negative fashion. This is normal.
Yes. I assume you are talking about the occasional image you see when a bright light or the sun shines into your eyes. If that's it you are experiencing a visual phenomenon called the purkinje tree. The purkinje tree is an image of the retinal blood vessels in one's own eye, first described by purkinje in 1823. This is a form of entropic phenomena. http://en.m.wikipedia.org/wiki/entoptic_phenomenon.