Is supercriticality dependent on partial presssure, or total pressure?
There is no such thing as supercritical properties. By looking at a gas under given pressure and temperature, you'll never be able to tell is it supercritical or not.

If you try to lower temperature or pressure and see what happens, then yes, the interaction is more complex, and a mixture of two compounds does indeed have a joint critical point different from those of both pure compounds; moreover, the very meaning of a critical point becomes different and more complicated.