Here are the mistakes identified in each of the provided sentences:

1. **Sentence:** Tropical forests exist close to the equator, **which** both high temperature and **abundant** rainfall occur all year round.
- **Mistake:** The word "which" should be replaced with "where" (or "in which") to correctly introduce the clause describing the location of the tropical forests. The sentence should be: "Tropical forests exist close to the equator, where both high temperature and abundant rainfall occur all year round."

2. **Sentence:** The Smiths are very proud **of** that their son always gets high marks **in** his **courses**.
- **Mistake:** The word "of" should be replaced with "of the fact that" to clarify the structure of the sentence. The corrected version should be: "The Smiths are very proud of the fact that their son always gets high marks in his courses."

3. **Sentence:** We called the baseball park **up** to ask **that** when the game was **scheduled** to begin that **afternoon**.
- **Mistake:** The word "that" should be replaced with "about" in this context to make the sentence grammatically correct. The corrected version: "We called the baseball park up to ask about when the game was scheduled to begin that afternoon."

4. **Sentence:** The American Indians **killed** the buffalo **only** when **necessity** to obtain food, **clothing** and shelter.
- **Mistake:** The word "necessity" should be changed to "there was necessity" or "out of necessity". The corrected version could be: "The American Indians killed the buffalo only when there was necessity to obtain food, clothing, and shelter."

5. **Sentence:** Some relatives **of mine** like **staying** at their cabin **on** Lake Omega every summer **holidays**.
- **Mistake:** "holidays" should be changed to "holiday" (or "during the holidays") for appropriate singular/plural usage in context. The corrected version: "Some relatives of mine like staying at their cabin on Lake Omega every summer holiday."

6. **Sentence:** In the United States **among** 60 percent **of the space** on the pages of newspapers **is reserved** for **advertising**.
- **Mistake:** The word "among" should be replaced with "approximately" or "about" to correctly modify "60 percent." The corrected version: "In the United States, approximately 60 percent of the space on the pages of newspapers is reserved for advertising."

7. **Sentence:** Gilbert Newton Lewis, **a** chemist, helped to **develop** the modern electron theory of valence, a theory **explains** the forces holding atoms **together** in molecules.
- **Mistake:** "a theory" should have a verb before it; "that explains" is needed. The corrected version: "Gilbert Newton Lewis, a chemist, helped to develop the modern electron theory of valence, a theory that explains the forces holding atoms together in molecules."

8. **Sentence:** Because of a high **birthrate** and **considered** immigration, the United States population **in** the late nineteenth century **increased** tremendously from 31 million in 1860 to 76 million in 1900.
- **Mistake:** The word "considered" should be "uncontrolled" or "high." The corrected version: "Because of a high birthrate and uncontrolled immigration, the United States population in the late nineteenth century increased tremendously from 31 million in 1860 to 76 million in 1900."

9. **Sentence:** For most of **their** history, **especially** since **the** 1860's, New York City has been undergoing major **ethnic population** changes.
- **Mistake:** "ethnic population changes" should be rephrased to "changes in its ethnic population" for clarity. The corrected version: "For most of their history, especially since the 1860's, New York City has been undergoing major changes in its ethnic population."

10. **Sentence:** Rocks **have forming**, **weaning away** and re-forming **ever since** the Earth **took** shape.
- **Mistake:** "have forming" should be "have been forming" and "weaning away" should be "weathering away." The corrected version: "Rocks have been forming, weathering away, and re-forming ever since the Earth took shape."