Do these two sentences have different meanings?
John has assumed a story.
John has done assuming a story.
If yes, what is the difference between the two? If no, can they be used interchangeably?
Neither of the example sentences means what the original poster intended. "Assuming a story" usually means "assuming a story (that someone else told the subject of the sentence) is true." "Being done" with "assuming a story" probably means "ceasing to believe that the story is true", but might mean "has finally figured out what the story is, so that he can assume the story is true."
So I will comment on the following four examples instead:
John has written a story.
John has done writing a story.
John has finished writing a story.
John is done writing a story.
Examples #3, #5, and #6 have similar meanings. Examples #5 and #6 are more specific than example #3. It is possible for "writing a story" to have multiple related meanings. Perhaps John outlines a story, writes a rough draft, edits the rough draft, and prints out the finished story. In this scenario, it is not clear whether example #3 means that John has written the rough draft, or written and edited the rough draft, or printed out the finished story. To me, examples #5 and #6 imply that John has finished the process, including printing out the finished story. Example #3 is very vague about how much time has passed since John wrote the story. Examples #5 and #6 imply that John recently finished writing the story.
Example #4 is not correct in Standard Written English. It sounds very similar to "John has done written a story", which is used in some non-standard dialects of American English.