Does the following sentence include a correct use of an appositive: "My teacher, Mr. Jones, is the best teacher in the world"?
Yes, Mr. Jones is the appositive of this sentence and it is correctly used to rename or further identify the noun: "teacher."
What edits should be made to the sentence "Their going home after school today" and why?
The sentence should read "They're (or they are) going home after school today" because "their" is possessive and "they" should be utilized as a noun.
In most interactions and communications with others what is, or what should be, the primary objective?
The primary objective in communicating with others is that there is a level of understanding of the information or theories you are attempting to transmit.