Answer:
1. HIS is correct. "He's" is a contraction of "he is."
2. HER is correct. "His" is a possessive pronoun that refers to a male. "Her" is a possessive pronoun that refers to a female.
3. ITS is correct. "It's" is a contraction of "it is."
4. ITS is correct. "It's" is a contraction of "it is."
5. HIS is correct. "He's" is a contraction of "he is."
6. THEY'RE is correct. "They're" is a contraction of "they are."
7. THEIR is correct. "They're" is a contraction of "they are."
8. CHILDREN'S is correct. "Childrens" is not a word.
9. MARY'S is correct. "Marys" is not a word.
10. HIS is correct. "He's" is a contraction of "he is."