star - Is the Sun homogeneous?

No it does not have the same composition everywhere. In the core hydrogen is fused into helium, so the fraction of hydrogen (denoted by $X$, between 0 and 1) decreases while the fraction of helium ($Y$) increases as time goes by. There is not much exchange of matter between core and envelope so the envelope will essentially have the same constitution as when the Sun formed.

In other stars the convection zone extends into the core, and for these stars there will be more exchange of the different elements within the star.

Evolved stars (e.g. red giant stars, horizontal branch stars) often have multiple shells where different nuclear fusion processes occur. An Asymptotic branch star, for instance, has a carbon-oxygen core surrounded by a helium burning shell, surrounded by a inert helium shell, surrounded by a hydrogen burning shell, surrounded by a very large envelope consisting mostly of hydrogen.

For non-degenerate matter density depends on the pressure. The deeper you descend into the star, the higher the pressure and therefore the higher the density.