1. Correct: The situation that he has to do better in school to keep his scholarship makes his mother cry.
Better: That he has to do better in school to keep his scholarship makes his mother cry.
Explanation: While the sentence is correct because "that he has to do better in school to keep his scholarship" renames "the situation," the sentence would be slightly clearer without having both a subject and an appositive.
2. Correct: Automobile manufacturers don't seem to care about my opinion that cars should all have huge sunroofs.
Explanation: "My opinion" and "that cars should all have huge sunroofs" are the same thing, so the latter acts as an appositive. In this case, the appositive is necessary to the meaning of the sentence, as the speaker could have many opinions about which automobile manufacturers do and don't care.
3. Wrong: The Halloween costume which he wore to the party left a trail of glitter.
Better: The Halloween costume, which he wore to the party, left a trail of glitter.
Explanation: "Which he wore to the party" is an inessential adjectival clause, so it should be separated from the sentence's essential elements with commas.
4. Correct: We don't quite believe her claim that everyone will make the team.
Explanation: She might have several claims, so the noun clause--"that everyone will make the team"--is an essential appositive.