Is there such a thing as a drainage survey, that would include evaluating a site's drainage issues, erosion issues, etc. Is there an American standard (i.e. ASTM) or similar?
A civil engineer will readily do what I've always seen called a "Stormwater Analysis". It will include any sub-topics you specify. I am not aware of a spec or standard for one, your contract will define the scope.
It will not include precise calculations for design of improvements. Those are done in "Stormwater Management Reports" and "Erosion & Sedimentation Control (E&S) Reports". It will include analysis upon which to base design strategy.
Remember: The Chinese ideogram for “crisis” is comprised of the characters for “danger” and “opportunity.”
-Steve