Star time

If I've understood the general theory of relativity, gravity "slows" light, and thus also time. So for a star, is there a significant difference in time difference between the core and the "surface" and does this have anything to say for a star's workings?

How do you measure/count time for a star? Does our system based on 133Caesium half-life work (months/years/millenia)?

Someone able to procure a formulae to be used?

I know one for newtonian gravitational field force, but not how acceleration effects time.

g=(γM)/r^2

If a photon were to travel from the core and outwards in a straight line, it would begin at c, then experience a linear decrease in speed (viewed from the earth) until it's out of the main body, it would then start to gain speed in a squared fashion until it reaches c again. Am I rite?