[boring serious post alert]
(1) Regardless of what my own views on the subject are (ambivalent, actually) I believe that before too long there will be a change in both the law and in accepted custom-and-practice regarding this. But more particularly I predict that not long after this change has been brought about, there will be an amazingly swift and almost unversal overnight change of public opinion. You read it here first.
Nearly everyone will suddenly say "but of course. What on Earth was all the fuss about? You wouldn't let an animal suffer pointlessly like that, and the more devoted you were to a pet, the more anxious you would be to let its life come to a dignified and painless end. Only a heartless sadist would want to do otherwise."
(2) The dangers, when someone is not demonstrably of sound mind and not able to communicate a clear, consistent, rational and informed request regarding their own death, are obvious and need not be repeated here. I freely admit, I do not know the answer to this question.
(3) There is a specific problem which is not always appreciated. A doctor, unlike a vet (who has had at least as much training) takes an oath (or at least, has a professional obligation) "to do no harm". (If I have got this wrong, I would be happy to be corrected by someone with appropriate professional knowledge.)
This includes taking life, which is why a vet can painlessly despatch a suffering animal but a doctor cannot deliberately cause someone's death. He is explicitly not allowed to, even if he wanted to, even if he thought it was the right thing to do and even if the law allowed it. So even a change in the law would not overcome his problem.
You will note that in American prisons, it is a prison officer and not a doctor who administers the fatal injection. I wonder if a new specialist profession will come into existance, the "humane human killer" who has all the medical knowledge neccessary but who is not actuall a doctor?
Obviously someone will have to come up with a nice name for this humane but thankless profession, which would stand alongside other highly-trained-specialists-who-are-not-actually-doctors, like radiographers and various therapists.
Diffcult questions with no easy answers, which I have first-hand experience of having watched over and cared for various much-loved relatives during the final stages of their lives. In my own case, I feel like Woody Allen - I'm not scared of death, but I don't want to be there when it happens.