1. Wrong: If you have any questions about the club, please ask Logan or myself.
Better: If you have any questions about the club, please ask Logan or me.
Explanation: As a direct object, the pronoun at the end of the sentence needs to be in the objective case. Myself is the reflexive case, which doesn't make sense here.
2. Wrong: You can give that candy to whoever you want.
Better: You can give that candy to whomever you want.
Explanation: As the object of the proposition, whomever is in the correct case.
3. Wrong: Me and Andy wanted to see that movie again, but we didn't know who to take with us.
Better: Andy and I wanted to see that movie again, but we didn't know who to take with us.
Explanation: The subject of the sentence should be in the nominative/subject case. Also, usually pronouns follow nouns and it's polite to put the speaker last.
4. Wrong: Joey gave the signed football to my sister and I.
Better: Joey gave the signed football to my sister and me.
Explanation: The final pronoun in this sentence is the object of a preposition.