Short answer: Maybe. Longer answer: This depends entirely on your locale and what kind of ticket it is. Parking tickets are commonly issued by what used to be called "meter maids." They are not police officers with powers of arrest. In some jurisdictions people can be deputized for a period of time or for a specific area like a college campus. While these individuals may not have full police authority (arrest, firearm, etc.) they may be able to cite someone for violations (speeding, disorderly conduct) and they may also have the authority to detain an individual pending a "real" police officer's arrival.
No, an off duty officer can not issue a ticket in the state of Alabama. An officer must be on duty before they arrest someone or issue a ticket.
No.
Yes.
Yes
no but he can issue you a ticket because of it.
Yes, the officer can write the ticket. It would be subject to the Statute of Limitations laws, but one month is not going to be an issue.
yes. they must show documentation of what the charges are.
Yes, in Indiana you can receive a seatbelt ticket if you are caught not wearing a seatbelt while driving, regardless of whether you are inside or outside of your vehicle. It is a primary offense, so law enforcement can cite you for not wearing a seatbelt even if you are not actively driving.
yes, unless he/she isn't driving a marked unit.
Yes, most definitiely.
As soon as the investigation is fully completed and he/she has determined fault.
If the officer who observes it believes that it is a distraction to your driving, he can issue a ticket for it and you can defend your actions in court.