Wow so many days for Tokyo! You can definitely do a lot and make many side trips. What are your interests for the trip though, any list of things to do or places to visit?
Will your group be using public transport, or driving? If public transport, JR pass for 2 weeks won’t have any cost savings since you’ll only need to use it on Day 2, 6/7, within city travel isn’t necessary. Even for JR 1 week Pass, you’ll barely hit the cost for a Tokyo-Kyoto round trip.
Kyoto to Kawaguchiko direct is not recommended, very long journey, and will need change of trains back in Tokyo, and another train down to Kawaguchiko. There are direct overnight buses though:
http://bus-en.fujikyu.co.jp/highway/detail/id/9/
Kyoto to Karuizawa is equally long, need to backtrack to Tokyo then travel up.
You are better off travelling to Kawaguchiko/Hakone from Tokyo. Maybe spend 2 or 3 days there, enjoy onsen and sightseeing.