Rare animal breeding

I think it depends on the breed or species. There are some mating rituals in which the male of the species is killed or otherwise dies in order to insure a healthier gene pool. It happens with insects like praying mantis, bees, maybe ants and termites too? Idk

But yea, as far as the rhinoceros goes, it would be a long incestuous process. It probably does happen in the wild and probably isn't the best option, making for a narrow gene pool. If it's inevitable, the best breeding scenario might be if the offspring breeds with its appropriate parent. That way the offspring of that union would only be 50% inbred instead of 100% like two siblings would be

Dog lines which have been "closely bred" is the reason some dog breeds have the weakenesses they possess.

Wider breeding and even making new crosses (like adding poodles + <whatever>) helps reinvigorate a breed / make a new, hopefully healthier breed.

yes and yes. populations can become small enough to effectively doom themselves.
the problems introduced with inbreeding tend to be overstated, but they do exist.

Yes it does cause problems like weaker immune system.