West Point grad and former faculty here (in their Physics/Nuclear Engineering department):
1. I doubt any POTUS has any meangingful influence on who decided to apply to a military academy. The application process take months if not more than a year, requires congressional nomination in most cases, etc. The incoming class to West Point is a reflection of those who desire to serve in the larger society. The students/cadets bring their values with them; most of them are solid and aspire to do the right thing. There are exceptions. There are extensive efforts at character education/development in general and honor specifically, but you are not going to change someone overnight, if ever.
2. I saw no correlation between honor cases and "religious loons". To be honest, I saw very, very few "religious loons" in the student population while on the faculty. Anecdotally, those being found on honor often had "co-morbidities" of discipline/breaking regulations issues where they were in trouble to begin with, but anecdotes are not data.
3. I think the "because they could" / temptation due to the virtual environment may be a factor. In a normal testing environment at USMA, there are other students in the room, though the desks are staggered to make it more difficult to accidentally see another's work. (The self-reporting sense of honor is strong -- cadets will turn themselves in because they mindlessly gazed at another's work while staring off into space during a test). But there is also the unspoken peer pressure to not cheat and the higher chance of being caught. (Interesting, faculty will often leave the room completely during an exam; our presumption is that cadets are honorable and do not need to be watched like a hawk.)
I tended to see more issues with honor in non-testing environments -- lab reports for instance. A single student has to make up a lab that he missed; he gets a copy from a friend "just to see how he laid out the report" and before you know it, they are misspelling "Newton" the same way in the same places in the report. Often there is a time pressure associated, and of course the grade pressure -- the cadet who gives in and cheats is often failing a class, close to failing another, and is already late on the report. They are human (as is the faculty and all of us).
The honor and character systems are designed to mentor those who can benefit from it and separate (expel) those who are too far along, are too grevious in offense, or have multiple offeneses. The mentoring process is no joke - it takes significant faculty time to do for any given cadet, not to mention the extra work placed on the cadet themselves.
The honor case, while run by cadet (seniors) with special training and experience in the Honor Code, is also facilitated by no-crud attorneys, and there are plenty of protections for the accused.
I do not know the specifics of how the cadets cheated and have no inside information about this particular event. I am curious.