Hi! That is a great question. With the three examples you have given, it is matter of understanding the use of indirect objects and indirect object pronouns. I hope the explanation below provides some clarification!
With the sentence, "Yo quiero que me traiga el coche a mí", both "me" and "a mí" acceptable to use since "a mí" is just used to add extra emphasis to the sentence. "Me" is used as an indirect object pronoun in this context because if you ask the question "I want you to bring WHO a car?" you can answer the by saying "I want you to bring ME a car", which the"me" in front of "traiga" answers. By adding "a mí", you are adding extra emphasis to the statement, but you could also just leave it as "Yo quiero que me traiga el coche."
The reason you shouldn't say "Yo quiero que me vengas", but rather "Yo quiero que vengas a mí" is because "me" is an indirect object pronoun and should only be used when an indirect object pronoun is required. With the verb "venir" (to come) you do not use an indirect object, because if you ask the question "I want you to come who?", you cannot answer with "I want you to come me." Rather, you would need to say "I want you to come TO me", which is why you could use "a mí" for extra emphasis instead of "me" (because there is no indirect object in this case). You could also just leave it as "Yo quiero que vengas" and it would have the same meaning as "Yo quiero que vengas a mí."
For the same reason you cannot use "me" in the sentence "Yo quiero que vengas a mí", you CAN use it in the sentence "Yo quiero que me cantes (la cancion)" Just ask yourself the question, "I want you to sing WHO (a song)?" and you can answer with, "I want you to sing ME (a song)" Therefore, me is the indirect object in this sentence, which allows us to use the indirect object pronoun "me" in front of cantes.