1) Does the author need to be an AZ state senator or representative?
2) does the same hold true in a presidential election, which one must the plaintiff be, a US senator, US representative or a presidential candidate?
Like the 2020 3rd party presidential candidate in WI or MI?
Only if the writ/case is regarding a violation of an Arizona election statute
2) does the same hold true in a presidential election, which one must the plaintiff be, a US senator, US representative or a presidential candidate?
No, Election procedural laws are the province of the individual states, so see No. 1 above