1) Linking Tatbeer to cupping is a misplaced link, because cupping is a medical treatment, while Tatbir is not. Hence, liking them is an attempt to find a way out of a problem faced by those performing Tatbir.
We can look at the issue in terms of jurisprudence from the point of view of the permissibility of self-injury and the impermissibility of it. The issue is controversial among the jurists, some of them are of the opinion that it is not permissible to inflict absolute harm to the body, and from here they issued a fatwa that Tatbeer is forbidden. While most of the jurists were of the view that it is permissible to inflict simple harm, which does not lead to serious harm to the body or may lead to self-destruction.
Hence, Tatbeer was considered to be of this kind, that is, one that does not inflict great harm on the body, and if a person deliberately inflicts severe harm on himself, he has committed a forbidden act, and he is a sinner.
This is for Tatbeer.
As for the example you gave with regard to eating clay, the example is not correct, because eating clay is absolutely forbidden, and it is permissible to eat a quantity of chickpea from the clay of the grave of Imam Hussein, peace be upon him only, and for the purpose of healing, not any clay.
2) As for the second question, this is a lie, as the Shiites do not say walking on hot coals and fire in the mourning of Imam Hussein, and this is an unacceptable innovation.
3) It is not an action of our deen, It is something outside of it, and after the explanation given, I need to say that S. Sistani says: If such act harms the mathhab, and reflects bad image, then it is forbidden.