If I could "guess". I'd say
he has a distinctive writing style, one that would be precise, to the point and listed. Think MLA format in college, something you learn your freshman year. If you're taking med classes you write distinctive in another fashion; APA format. It's very hard to break that habit when talking to people online. I'm sure anyone he's spoken to online noted how rigid his interactions were.
I'm guess him to be heavily involved in her "search / rescue", most likely online.
I do not think a person with schizo-affective disorders would be able to be this cunning. I speak from experience in the field. If you look at cases of dismemberment and this disorder, the person who does it - is dazed and doesn't know to clean up or behave in a manner to hide their crime. It happens during a psychosis. That maybe hard for people to understand in the sense that a psychotic would only do something like this. But being psychotic doesn't mean a person has a chemical imbalance.
(hope that made some sense).
This is all too much like the film "We need to talk about Kevin" and I highly recommend everyone read the book, wiki it or look on Youtube. This might help people understand the minds of a sociopath, which is the type that does this sort of thing.
He had an agenda and probably had access to a lab, given his academics and right now is midterms. People are on campus at all odd hours (Im currently a med student so I see people at 4am cramming for studies). Seeing him around wouldn't be odd this time of year. I mean, at least I wouldn't think a kid who's trying to be the next Doogie Houser would be odd spending days off at school. And I believe in the time she was missing, his school calendar might reflect a fall break or some other break. I wasn't out for Columbus day but our fall break was the next week. Something to think about.
He's disciplined, brazen and most likely the next Ted Bundy but stopped. The only reason that jogger escaped is because he doesn't have the gull to really push with an adult, she was a threat to him in the end but a child? Easier. (JMO). This also explains the chemicals, he most likely used embalming fluid and that would be the "smell".
This is all me speculating and I do not think this is his first kill, however, if there are others they would be near by. I would think it's mostly animals though, ones he's dissected to learn. I know people are hung up on the cross bit (no pun intended there), he's at an age where most of us were looking into all sorts of faiths and trying to learn. I know at that age I was curious about everything the world held. He might have had a faith that was a mixture of two. Say... Jewish/Christian. *I'm a Jew, and there's reasons why I'm bringing up that part*. His fixation it seemed with #'s, relation to 3 and maybe even his online names will give away his inspiration and where his head is at. See, at that age he THINKS he knows everything but he has no experience as a criminal or how to survive much. He did live at home, so I bet he left an online trail. He's too analytical, meticulous, daring and curious to just abandon her case and not try to help. I bet he offered help in all sorts of ways, tossing out "idea" of the killer, but not exposing who he was directly. If he was doing that then I'm sure he also did red herrings and pushed false information and leads that were debunked, to thwart the conversation but still keep engaged.
If they look at his online activity, I'm sure a lot will be answered.
BUT THIS IS JUST MY OPNION and I'm not LE, or anything. Just an observer.