I take it this is a tour-group rather than something you intend to DIY? Because the language barrier can make DIY travelling around really quite challenging at times.
Either way I get a sense that the itinerary is trying to fit too much in, for example 5 days in Tokyo (does that incl the day you fly in?) but with three side-trips planned – why? You’d be leaving Tokyo barely having seen anything, and it alone has sooo much to offer... hmmm.
Nikko – Haven’t been.
Mt Fuji – I went there to climb it which was a memorable experience

Check the climbing vs closed seasons if you’re considering climbing, even in summer it has a snowcap and can be a mighty challenge. Also bear in mind the notion that you ‘can’t see a mountain when you’re standing right next to it’. On a clear day you can see Fuji-san from Tokyo, and it’s c60 miles away! Now that’s genuinely awesome in itself, and the views could/should be better from closer but not too-close towns. So I’d suggest if not climbing, then you need a compelling reason to visit the town itself.
Kamakura – HB, wish I’d have had time, it is considered one of the best castles in JP.
The only other city I’ve visited on your list is Kyoto which was amazing. Pick a couple of temples/gardens, don’t attempt too many and end up getting ‘templed-out’

Maybe remind yourself that especially the temple gardens are meant to be contemplated, and to draw what’s intended out of them, in the same way as great and complex painting, it requires time.