Strictly speaking, it must give the _gene_ a reproductive advantage, not necessarily the animal. Often, these two advantages are aligned. But they need not be. This is the case in kin altruism. And this is also, why "promoting longevity beyond the reproductive stage of one's life cycle" affords sometimes very much such benefits. Example: grandparents can help raising their children's children (and their children's children's children ;)). Kin altruism becomes even more apparent in ants (or other social animals). Worker ants or bees are pretty much infertile but they help a lot.

> To survive natural selection the gene must give the animal some reproductive advantage.

Most probably, but not necessarily.

A gene can be present without being useful because it is linked to another gene that is itself useful; or it can be a leftover from an earlier moment of evolution where it was useful, and then it ceased to be useful but there was never an evolution pressure strong enough to have it disappear.

In summary, just because a gene is present and active does not mean it's useful today.

Not necessarily. The mutation could have survived for any number of reasons including genetic drift. As long as it didn't kill the organism right away there wouldn't be significant negative pressure to get rid of it.

Perhaps not having the ability to regenerate leads to more cautious behaviour, avoiding life threatening situations to a much greater extent? Puzzling...