Question: A block is suspended from a string; does the gravitational force do any work on it?

Answer: 
Work is performed when the point of application of a force travels through a distance in the direction of the force. 

An object suspended by a string in a gravitational field experiences the force of gravity but if it does not move, then no work is being done on it.
No, if the block is still, there is no work being done on it. There is a force on it, yes, but no work. Work is really a function of a period of time i.e. in 1 second, it did 1 Joule of work. It does not make sense to ask the value of such a quantity for any given moment.