During CIVIL WAR, several superheroes and criminals who were not American citizens were deported rather than imprisoned in the Negative Zone. The SUPER HEROES of EUROPE, or S.H.E., may have diplomatic immunity. During CW, they had a meeting (where Norman Osborn, under orders from Stark, organized an attack on an Atlantian official) and some NYPD officers grumbled how "their" heroes had to register, but superheroes from Europe (and presumably other nations) didn't.
Granted, other countries, especially Canada, Japan, and Russia, had some version of a SHRA (with the former Soviet Union being the most restrictive about it, of course).
But, yeah, it does depend on writer whim, of course.
As for supernatural realms, as the example cited of Satan (or any one of the Hell-Lords like Mephisto or Lucifer or Marduk Kurios, who rule other dimensions considered "Hell"), I doubt they would be free from arrest under any sort of diplomatic immunity. Granted, it would probably be impossible to "arrest" most demons or mythological threats anyway. Usually you either slay them, or they escape, and can often be reborn through arcane means. Considering how much Tony Stark (who runs SHIELD) and other feds "hate magic", I doubt they would allow diplomatic immunity to any mythical being. Thor had to basically pummel Iron Man and threaten Washington with a thunder-strike to get considered "neutral" to the SHRA. Hercules is a god from Olympus but he is still considered an unregistered fugitive. If given half the chance, I am sure the feds would go after Ghost Rider. You could even stretch and claim that Moon Knight is a "diplomat" working for an Egyptian God, but that didn't work and the feds are after his wack-job butt.