I think you must be having problems identifying which noun in each sentence needs to be made possessive. This can be determined by thinking about the meaning of the sentence.
In your first sentence, an umbrella cannot own anything; it must be that the umbrella belongs to Peter. To make the noun (and proper name) Peter possessive, add 's--Peter's umbrella is gray.
Cindy does not come from a story--the story comes from Cindy, so the sentence could read: That is Cindy's story.
In the third sentence, we hope that the eyes are green instead of Peter being green! Since the eyes belong to Peter, your sentence would read: Peter's eyes are green.
In your fourth sentence, you would add 's to Sandy (the owner of the car) and the sentence would be: This is Sandy's car.
You would not change the "y" in Sandy to an "i" and add "es" because you are using the apostrophe to indicate possession of the car by Sandy; you are not indicating that there are more than one Sandy--"Sandies" would be a group of something all called Sandy.