Gnarly PDEs aren't exactly the same beast as differentiating single variable polynomials.

Gauchy theorem and so

Probably infinite series? Maybe differential equations, but I guess that's usually its own subject.

I would think at most institutions it would be calc 3 and beyond, or pretty close to?

A lot of majors don't take beyond 1 or 2.

Nothing. It's a meaningless term. Some people use it to say "hard calculus" (multivariate, pdes, etc) and I suspect that's how it's being used here but, historically, it has mostly been used to refer to the introduction to real analysis courses found in books like Spivak and Apostol.