Bless him, hope he's feeling a little better tomorrow!
I accurately remember round the same age, I was diagnosed age 12- feeling rough and had headaches for a while after beginning insulin treatment- had ketones present- a number of sick bay visits. Maybe his body will take time to adjust to the insulin regime! Things are probably a bit different now, but I remember a lot of teachers not understanding diabetes what so ever, I had one teacher refuse to let me each something sugary in a lesson because her "nan is diabetic and she doesn't need to"..
I'm sure things are different now with better awareness and increased number of children diagnosed with T1, but make sure they're plenty aware.
If I could go back to those days, one thing I'd change is eating more regularly and routinely whilst at school, as due to the timetable eating a big lunch at a set time once in the day didn't work well for me (caused me to yo-yo) eating less and more regularly throughout the day would have made things better I reckon.
I used to get bad headaches at that age when high OR low, but I only really get them these days if I've had a serious hypo.
I know how difficult it must be at his age to make him to do things! But encourage him to drink much more water than usual to flush anything out that could be floating around (which would probably be a good idea if he's coming down with something also). If he is coming down with illness be aware as i'm sure you know that his levels will be much more sensitive.
I could be wrong here, but something I found gave and does give me headaches is eating white bread/pasta which I assume is caused by it being fast release and converted into sugar very quickly- (just remembered this from eating white bread rolls at school!) Potatoes do something similar, although everyone is different.
"Should I make him eat something even though he feels ill? Should I just stop thinking about it all so much?"
Just make sure that before bedtimes especially that he has eaten something, hunger can be a symptom of both high and low blood sugars- speak to doc about insulin requirements/dietary requirements when ill.
I hope you're coping okay! It's a crazy thing to get your head around, but it'll be crystal in no time I'm certain. He will instinctively understand what his body is telling him over time and how it reacts and what to do etc!
Hope you're well
All the best