1) He was asked to solve a mystery on Peter Williams's farm (is on correct) because his ring (is it clear it is Mr Williams's?)had been stolen from his bedroom

2)She sat down in an empty seat (or on an empty seat) and a man asked to leave him the bus seat (asked her to leave her bus seat to him?)
3) he wanted that one day blacks and whites could be friends (he wanted blacks and whites to be friends one day?)
4) Can you start by saying: "It's a person who only thinks about himself?"
Thanks

1. Peter Williams' farm (no extra -s needed on a multi-syllable word). No, it's not clear which pronouns refer to "He" and which to "Peter Williams'"

2. She sat down in an empty seat, and a man asked her to give up her seat to him.

3. In ( ) is correct.

4. Should be "... a person who thinks only about himself." You want the word "only" as close as possible to whatever it's modifying.

1) Yes, "on" is correct in this sentence to show the location of the farm. However, it is not clear whose ring was stolen. To clarify that it was Mr. Williams's ring, the sentence should be modified as follows: "He was asked to solve a mystery on Peter Williams's farm because Mr. Williams's ring had been stolen from his bedroom."

2) Both "in" and "on" can be used in this sentence, but "in" is more commonly used to indicate the action of sitting down. The sentence can be revised as: "She sat down in an empty seat, and a man asked her to leave her bus seat for him."

3) The sentence is grammatically correct, but to make the meaning more clear, it can be revised as: "He wanted blacks and whites to be friends one day."

4) "It's a person who only thinks about himself."