Which distributions have polkit installed by default?

Most that come with a desktop environment. (idk if the non-systemd versions are vulnerable though)

Which applications need polkit?

If you get a password prompt when a program needs higher privileges it most likely uses polkit.

Do you have polkit installed?

Nope, I just couldn’t get it to work :D It’s pretty annoying because some programs don’t work without it, but most of the time you shouldn’t be running those anyway.