A continuous center point of a convex spherical polygon
For your second question...

For any distribution, if you take a sample $R$ of size $O(1/epsilon^2 )$ of it, and compute its center point, it is going to be a $\geq (1/3-\epsilon)$-center point, with probability $\geq 1-\epsilon^{O(1)}$ by the $\epsilon$-net theorem (well, more precisely the $\epsilon$-sample theorem). You can apply this to a polygon - you just need to sample points uniformly inside the polygon. This works in any constant dimension.

Similar ideas were used in the paper "Approximating Center Points with Iterated Radon Point". Or so I believe... ;)

Computing the exact center point for a general polygon is nasty as far as I know. There was a paper in the 90s by Dan Halperin on maintaining the area bisector of a polygon, and it was painful, if my memory serves me right.