If parts of the universe are accelerating faster than the speed of light does that mean that stars can catch up with and overtake the light photons that they have previously emitted traveling in the same direction that they are being expanded in? Or is it that just the space between stars is being stretched to the effect that the light/photons they emit become trapped in that ever stretching space and can never escape or be overtaken by their own star? Also, if a star can intercept with its own photons what occurs when it collides with them?
No, that would not happen. The expansion of space accumulates over distance. The expansion to exceed speed of light has to accumulate over very large distance. Moreover, there is no one direction of expansion of space. It is expanding in every direction and that is how it accumulates over distance.
If the light is moving in the same direction in the neighborhood of a star, it is going to pass the star because, the expansion is not in only that direction alone. And the expansion rate would be slow between light and nearby star. Same way, there is no point of taking over as the space is expanding in reverse direction as well.
To understand it, consider a big inflating balloon on which an ant is crawling. The balloon being universe and the ant being light. If you consider two distant spots on the balloon, the distance between them can increase at a faster rate than the speed of ant on the balloon. But any spot (star) on the balloon can not catch up to the ant let alone the over taking in whatever direction the ant moves.
On scales much smaller than the Hubble length $(c/H_0)\approx 4 Gpc$ one can use the Newtonian approximation for the expansion and define the expansion velocity as $v=H_0 d$, where $H_0$ is the Hubble constant, $d$ is the distance, and $c$ is the speed of light. It follows from these formulas that where this approximation can be used, $v\ll c$. For larger distances, general relativity is needed to model the expansion and relative velocity is not well defined for two separate spacetime points in general relativity. So it doesn't make sense to say "the universe is expanding faster than the speed of light". Therefore nothing can overtake light even in an expanding Universe.