EDIT: I should clarify that there’s not one vegan philosophy. There’s many different philosophies that could lead to veganism. Animal personhood being the most extreme end of it, but vegans also include people who believe in harm minimization, people who just hate factory farms and live in cities, Buddhists, radical interpretations of halal, and more. I answered these questions from a harm minimization perspective.

General principle is minimization of harm. The classic example is “You’re on an island alone, slowly starving to death. There’s a pig. Would you kill and eat the pig?”

For quite a few vegans, the answer is yes. Luckily, that’s not the situation we find ourselves in, we can live healthy and happy lives without harming many animals in the vast majority of situations.

To directly answer the question: it depends. Is there an alternative that hasn’t been tested on animals? Is this medicine life-saving, or just very slight quality of life bump, like getting over a hangover slightly faster? Those questions would guide you to an answer.

To answer your chicken question, I don’t think there’s any moral issue with eating the body of a being that’s died of old age. I don’t think many vegans would do that anyway though, because after a long time without meat, it tastes “wrong” to eat meat (not sure exactly how to describe it). Same reason not many long-term vegans are that interested in lab-grown meat.