Should I go with my father to court to testify that he is alive and that he is my father, despite my mother's objection, given that I know he is alive and I continue to spend his pension money?

Your question is unclear. It is not clear to us how your mother was divorced from your father, nor whether it was your father who divorced her or the judge. We also do not understand what is meant by your father wanting you to testify that he is alive. Such matters belong in a Sharia (definition) court, so the issue should be presented to the court. It is not permissible for you to obey your mother in concealing truthful testimony. And Allah knows best.