Locate the gerund and identify its use. he wrote a book on the fine points of running. subject direct object predicate noun object of a preposition

The gerund in the given sentence is the word "running".  The gerund word is used as an object of a preposition. A gerund is a word that takes the verb form in -ing, but functions as a noun in the sentence. The word "running" is used as an object of the preposition because it is preceded by the preposition "of". 

The answer is object of the preposition. This is 100% correct.

The other animals thought that the cat was arrogant, superior and possessed a petulant independence.

Explanation:

"The cat walked by himself" tells the story of how human beings domesticated all animals, but he had trouble taming the cat that refused to submit to human wishes. The cat was also not very popular with the other animals, who were offended by the cat's behavior, who were always very arrogant, with an air of superiority and petulance.