The correct answers are The soup I ordered was completely cold. and The waiter, who dropped my pizza, didn't even apologize.

The soup I ordered... is correct. In a defining relative clause, you can omit the relative pronoun (that/which) if it is the object of the clause (I ordered the soup).

The waiter, who dropped my pizza... is correct. It uses a non-defining relative clause properly separated by commas.

The chef, I spoke to earlier... is incorrect. You cannot omit the relative pronoun in a non-defining relative clause. It must be "The chef, whom/who I spoke to earlier, was very rude."

The dessert that it arrived last... is incorrect. The relative pronoun "that" replaces the subject "it", so you cannot use both ("that it"). It should be "The dessert that arrived last..."