If someone is waiting for me on the ground floor is it valid to say " He is waiting for me downstairs" or could we just say " He is waiting down". Does 'waiting down' has different meaning?
"He is waiting down" is not a usual expression for anything. If it were used, it might mean that the person is somehow trying to out wait another person, or some sort of deadline. It does not mean "He is waiting on a lower floor".
He is waiting downstairs" is a very normal and usual expression when someone is waiting on a lower flower. 'downstairs' is the most usual way of referring to the location on a lower floor from where the speaker is.
The phrase "down the street" refers to a location some way along the street, and can be used to describe events. Similar forms such as "down the path", "down the beach", "down the hall" are common also.