So they actually have absolute authority? Like if they decided they wanted to execute someone for being gay or a heretic could they?

(Not that the contemporary Catholic Church advocates or would do that. I’m asking a legal question.)

I’m really not sure. Wikipedia lists them as a “Unitary theocratic Catholic elective absolute monarchy” and says “ It became independent from Italy in 1929 with the Lateran Treaty, and it is a distinct territory under ‘full ownership, exclusive dominion, and sovereign authority and jurisdiction’ of the Holy See, itself a sovereign entity under international law.”