In this excerpt, the meaning to the word tempers is to exercise control over, which, according to the Merriam-Webster Dictionary, is an archaic use of the word. In fact, there are other indications of somebody exercising control over somebody else in the passage, for example, "men are ruled by women" "Was it not she (...) That made him send Lord Hastings to the Tower (...)?"
As regards suppliant, in this excerpt it means a person who respectfully asks or begs something to someone in power or authority. In this case, Lord Hastings was a suppliant to the Queen for his delivery.
