Here are the mistakes underlined with their corrections:

1. If I have time after work, I will go for a walk to relax. _(No correction needed)_
2. Should he **exercise** more often, he will be healthier.
3. Unless she doesn’t stop eating junk food, she will gain weight. _(No correction needed)_
4. I would go to the gym tomorrow if I have time. _(Correct "gym'" to "gym")_
5. My friend’s children are my adopted nephews and nieces.
6. Can you help me with my homework, please?
7. I am a big fan of yoga because it helps me relax.
8. The emperor would bow for his subjects as a sign of respect. _(Correct "bow" to "would bow")_
9. In case it rains, don’t forget to bring your umbrella.
10. Should you **study** more, you will pass the exam with flying colours.