Tokyo, Nikko, Kyoto, Nara in 6 days

We are travelling to Tokyo from March 31-April 5. We are staying in Tokyo for the first two days. Out of those we will visit half day or so to Nikko. Then going to Kyoto by train late evening on April 2. Stay in Kyoto for two nights. Out of that half day go to Nara. Our flight is April 6 early morning at 12:30 am from HND. I have a few questions:
Is there any guide service in Nikko?
How to go to public hot springs near Nikko? Or is there any hot springs for public near Kyoto/Nara? What would be the best way to do this?
We are thinking of buying JR pass but Tobu line makes sense for Nikko so not sure what to do.
Since we are only staying 6 days and want to make the most of our trip, is it a good idea to buy JR pass? Some of the bullet train does not accept that and we do want to get to Kyoto and Nikko without wasting too much time on the train. Not sure how to do this.
We also want to visit Mt. Fuji but not sure how to fit that in our itinerary.
Any suggestion would be very helpful.
Thank you,

It will be difficult to do a half day in Nara, and even more difficult to do a half day in Nikko.

- I remember seeing a free "volunteer" guide once you are in central Nikko area. They give you full explanations about the temples/shrines. They don't help you rush from one temple to another to cover many places, though.
- Visiting Nikko from Tokyo takes a full day; you cannot go and come back in half a day.
- If you are making a round trip between Tokyo and Kyoto (I assume you are arriving in Haneda and also departing from Haneda), then a 7-day JR Pass makes sense. You can use this to go to Nikko as well.
- You "might" be able to see Mt. Fuji from the Shinkansen between Tokyo and Kyoto. WIth only 6 days, I would not try to fit in a "visit" there.

What time do you arrive on March 31st?