1. Wrong: Either Robin or our coaches together with the baseball team goes to Deano's after the game.
Better: Either Robin or our coaches together with the baseball team go to Deano's after the game.
Explanation: the verb should agree, in either/or sentences, with the closer subject. Together with the baseball team is a prepositional phrase, and therefore has no effect on the verb.
2. Wrong: A small percentage of the students leave each weekend for sporting events.
Better: A small percentage of the students leaves each weekend for sporting events.
Explanation: A small percentage is a singular subject. Of the students is a prepositional phrase.
3. Wrong: Many editors of The Record has won awards for great journalism.
Better: Many editors of The Record have won awards for great journalism.
Explanation: The plural editors is the subject of this sentence.
4. Wrong: After hours of deliberation, the jury doesn't agree with each other about a verdict.
Better: After hours of deliberation, the jury don't agree with each other about the verdict.
Explanation: While jury can be singular or plural, here its meaning is clearly plural.