Can the school punish one or both students for their conduct?
I don't know. Can they?
May the school punish one or both students for their conduct?
Sorry, couldn't resist a little humor. They are certainly able to discipline to the extend they can. As this is not a court system so the burden of proof need not be as strong as if this was a criminal case. The best course of action for the school will likely to refer the accusation of a crime to the police who are better equipped to investigate and determine if any criminal action occurred. As for what will happen while the criminal investigation is being made.
If so, would they stand a chance of challenging the ruling in court based on how the Supreme Court's previous ruling?
Based on your search? Likely. Whatever communication made was made between the two young ex-lovers, the school could find no evidence of it traveling through school resources so your search will not be the at issue element of any challenge, and rather it will be the decision occurring in the absence of evidence or the basis that evidence gathered was not in school custody and could be altered by one party that provided their own copy of the transcripts. Either way, parties can file sue for any reason and it's up to the courts if the reason is valid before discovery and trial.
Would the punishment(s) and/or their chances of a successful appeal following one vary significantly had it been an offline school day or had I found offending logs in the school server?
Likely different, as I am not entirely convinced that the disruption of the classroom occurred sufficient by the involved parties to cause a distraction to an uninvolved third party. As such, the mere fact that they were texting or passing notes (like the good ol' days) would be more likely to disrupt as third party students would be involved. That it was not done in a public way that was visible to the class writ large or recorded in the logs suggests the distraction did not occur, as no one was aware of the action at the time it happened. Had this occured in a physical classroom, the teacher and remaining students would be able to recall the disruption occurring to some degree and not rely on the School Resource Officer to check the security cameras to confirm that it took place (That isn't to say they won't be asked to pull the records, but that they would not be needed to prove or disprove the alleged accusation as a disruption.). The teacher would have grabbed the passed note, or noticed the cellphones and engaged in immediate disciple actions OR the students would recall Kevin asking them to pass a note to Winnie Cooper or watching Kevin and Winnie texting with their phones under their desks to avoid getting noticed.
And obviously, had the logs been found to support the interaction took place (whether it was criminal or not) while using school resources, then it is likely that the disruption part of the law does come into play.). At either rate, most computer systems that have public use accounts such as school computers will have a disclaimer that use of the resource by the user consents to monitoring of all activity while using the device or service, which means that there is an argument that they waved their rights against a search and seizure of the electronic traffic on the school's network, systems, hardware, and other such IT infrastructure the moment they put in their username/password and hit enter, and that such communications are then subject to the rules of the IT policy of the school and they can be disciplined for misuse of school property if it rises to that level. If there was evidence of a crime, copies of those transcripts would be useful in getting warrants to have service providers used turn over any transcripts to the investigation.