Is "Al-Qassam" one of Allah's Beautiful Names or Attributes? If not, what is meant by it, and what is the ruling on saying, "I will hit your Qassam" or "Leave me out of it"?

There is no evidence that "al-Qassam" (the Divider/Distributor) is a name or attribute of Allah, nor has it been counted as such. Rather, it has been reported that Allah divides, as in His saying, the Most High: (It is We who have apportioned among them their livelihood in the life of this world), and His saying in the Hadith Qudsi: "I have divided the prayer between Myself and My servant into two halves."

There is no harm in reporting about Allah that He "divides/appportions," because the scope of reporting is wider than the scope of names and attributes. What is attributed to Him, the Most High, in the context of reporting does not require a divine textual basis, unlike names and attributes.

As for the phrases "I will strike your qassam," or "leave me from him," no ruling can be made on them without knowing their meaning.