The word "grate" has multiple valid meanings depending on context:
As a verb, it means to shred into small pieces (especially food items like cheese).
As a verb, it can also mean to irritate or annoy someone.
As a noun, it refers to a framework of metal bars, typically in a fireplace or drain.
Sentence I: Correct - "grate the cheese" is the proper use of "grate" as a verb meaning to shred.
Sentence II: Correct - "grate on his nerves" is also correct, meaning to irritate or annoy.
Sentence III: Correct - "fireplace grate" is a valid noun usage referring to the metal framework in a fireplace.
Hence, the correct answer is option (e): All I, II, and III.