What do you mean, for what practical implication I’m asking? And I don’t understand what it means that it’s a matter of definition.
Could you explain, so that I can define the questions better?

Think about each of the questions and you’ll see that it’s just a matter of definition. The first is basically the Ship of Theseus, where all the planks are replaced and the question is when it stops being the same ship. There is no way to answer that. You can arbitrarily define whatever point you want as the point where it stops. And so too with the other two questions. Is a person the combination of body and soul or not? Define it however you want.
If you present the question regarding some specific implication, like for example from what point it is permitted to kill him because he is no longer a person, or something like that, then perhaps one can try to think about it. And even then, my estimate is that the laws of doubt would apply there.

Why, in your opinion, is there no principled way to answer these questions? What interests me is the metaphysical question, that is, understanding when I would say about something (material or conscious) that it has lost its identity. Still, here are a few formulations I thought of:
1. I’ll actually start with the third section—suppose I have two friends, one of whom had an accident and lost a significant part of his body (for example, everything not essential for the body to continue functioning—legs, arms, a kidney, and so on, though of course not the brain), and the other experienced a psychotic episode and is “no longer himself,” meaning his whole personality changed and he no longer behaves rationally (you could say that his soul has changed almost completely). Now, we went out together, they are in a car and I am on a motorcycle, and they had an accident (poor guys, I almost killed them with my story). I have the option of taking only one of them on my motorcycle to the hospital because their car is wrecked, and we are far from any help so I am the only one who can help. Is there any priority to saving one of them? Or does my obligation to help apply to both in exactly the same way, with no priority to either one?
2. A state of dissociative fugue is a state in which a person completely disconnects from his identity and replaces it with another identity, but only for a certain period, after which he returns to his previous identity. If during the fugue the person committed a crime, and by the time they come to convict him he has returned to being himself (I am assuming here that he did not fake the whole story, which is what research generally assumes), is there room to convict him? After all, he was not coerced, and even as a person with another identity he had free choice. But at the moment one could argue, in a double sense, that he did not commit the crime but rather a different person did.

I explained why it can’t be answered. I don’t know what I’m supposed to add.
In the examples you brought, you yourself demonstrated how resorting to a practical question makes the discussion possible. But as you’ll see, the discussion does not deal with the metaphysical questions you raised.
1. That would depend on the laws of priority in rescue (and not on the metaphysical question). Clearly there is priority for the one who is ill in his body and not in his mind, since priority in rescue is determined by the number of commandments a person can fulfill. The one who is mentally ill is exempt from the commandments. If we are talking about an illness in which the personality has changed but he is still of sound mind, then in my opinion they are equivalent. One can make expectancy calculations—how long they will live and how many commandments they will fulfill.
2. That would depend on your theory of punishment (and not on the metaphysical question). If the purpose of punishment is to deter society, then certainly he should be punished. If the purpose is also to deter him himself, then it would also seem that he should be punished so that if he returns to such a fugue he will not sin again. If there is no such concern, then no. If the purpose is to protect the environment from him, again there is concern about a recurring fugue. If there is no concern about a recurring fugue, then there is no need to punish him. And so on.

I understand your point, so I’ll leave the metaphysical question aside.
1. If we leave the halakhic consideration aside for a moment and speak on the moral plane, what would the consideration then be? It is hard to think of a utilitarian consideration here, since both of them experience happiness and pleasure (in people with schizophrenia—they usually do not express emotions, but presumably they do experience them). Would the consideration be the number of good deeds each of them will be able to do? What other considerations would there be?
2. The second purpose is not relevant because it is rare (if it happens at all) for someone to return to fugue. As for the first purpose, it is definitely relevant. But I am again redirecting the discussion to the moral plane: should such a person be condemned? For example, if he is my friend, and the crime he committed was murder, and I assume I have negative feelings toward him, is it proper to cut off ties with him? Is it proper for the social environment to condemn him and exclude him from social activities?

1. I already answered that.
2. No. If his consciousness is completely different, then why should you condemn him? It’s like a penitent who says, “I am not the same person.”