The correct answers are My nephew, who lives in London, will inherit the mansion. and The diamond necklace that is hidden in the safe goes to my cat.

My nephew, who lives in London, will inherit the mansion. is correct. It uses a non-defining relative clause (enclosed in commas) to give extra information about the nephew.

The diamond necklace that is hidden in the safe goes to my cat. is correct. It uses a defining relative clause (no commas) with the pronoun "that" to specify exactly which necklace is meant.

My favorite painting, that hangs in the hall... is incorrect because you cannot use "that" in a non-defining relative clause. It must be "which."

My husband, whom I met in Paris will receive... is incorrect because it is missing the closing comma after "Paris." Non-defining clauses in the middle of a sentence must be enclosed by two commas.