I know that Postal Service was authorized in the Constitution, but I'm confused about its status in the here and now.

It's considered a government agency, but gets no taxpayer dollars. So in a way it's independent of the rest of the U.S. Government. The Post Master General is not re-appointed every time a new President takes office - sort of like the Chairman of the Federal Reserve.

Does Congress have any input into the day-to-day operation?