Hi, I was a page 2 years ago. Obviously if you end up doing summer, school is not an issue 
I was a Fall page and I must say that from my experience and hearing summer page experiences, I highly recommend going for the semester program. You’re there for much longer, have so many more experiences, and build so many more relationships.
Classes are offered in the 4 major fields of English, Social Science, Physical Science, and Mathematics. The teachers generally teach to AP guidelines and if they’re not, they can usually still help you out with being prepared to go back into your class/ for completing the AP exam (depending on Fall or Spring semester). Calculus is offered, Physics is offered, you could choose between American Literature and English Composition. Unless they’ve changed something World History is not offered. The options my semester were US History or Political Science.
It really depends on how your home school ends up transferring credits back. All of my classmates had different experiences with that. I will tell you that the classes were some of the most difficult I’ve ever taken. Even if you don’t find the material too difficult, the fact that you’re only going to classes for 3 hours a day at the most make things very unusual. There’s a high amount of homework that has to be done usually while multitasking with Senate responsibilities.
Inauguration will be somewhere in the middle of the Fall and Spring. Not sure which will get to see it! It’s right on the cusp!! Many times it won’t be up to you, however. For example, my sponsoring senator appoints males in the Fall and females in the Spring, so I had to do fall despite the fact the Spring would’ve been more convenient for me.
I will be rather blunt with you regarding extracurriculars. Yes, obviously they make your resume look good. However, if you’re that concerned about missing them, then perhaps this program isn’t for you. The program requires you give up a lot. For example, because of the way my school transferred grades back, for me, it ended up meaning giving up my top 5% ranking in my graduating class. However, this is a program that only 60 teens your age in the whole country will get to do (I don’t count summer because the program is so different). It’s a once in a lifetime experience in my opinion and I certainly wouldn’t have given up the experience for anything even though I had to give up a lot back home!
If you have any other questions, please feel free to ask or PM!