Ok, I will try and provide some help based on my very similar itinerary in October (Nights 1-8 Tokyo, Night 9 Hakone, Night 10-12 Kyoto, Night 13 Mt Koya, Night 14-15 Tokyo then home).

Tokyo - I can't comment on Nikko or Disney Sea, as I did neither. I did try and go to the fish market, but our self proclaimed expert got the times wrong, so we missed the auction. We did get sushi at around 4am from one place, and I am told it was very good (I don't eat meat or fish). There are thousands of food options for those who eat meat, and almost as many for those who eat fish, especially if she is ok with fish broth / fish flakes that you may not even notice. I don't eat any fish at all, so my options were much more limited. The little 'hole in the wall places did great food according to my omnivore friends, everything from sushi to tempura to noodle places, at every price point. I found the department store basements fascinating food wise. If you post where you are staying I am sure people can recommend good restaurants in the area.

Hakkone - I stayed in a ryokan and really enjoyed the experience, although in retrospect it was probably overpriced for what it was. Dinner was included, and the lovely concierge at my Tokyo hotel called the ryokan for me and explained my dietary restrictions, and they were wonderful about making changes (this is not the norm I am told). The dinenr had about 20 different dishes, it was a real adventure, some was great (amazing tofu) some not so much. Breakfast there was the choice of western (eggs and toast) or Japanese. There were shared indoor and outdoor hot spring baths, and also a private bath you could book out as family / friends. Loved having a nice hot soak. Our ryokan also had a private bathroom in the accommodation, but many do not.

The concierge also wrote a card out explaining what I did and did not eat, that I could show restaurants, and I learnt the Japanese phrases as well.

We took the shinkansen from Tokyo to Odawara, then the train into Hakone, left our bags with the ryokan then hopped back on the train and did the Hakone loop (train, cable car, rope way, boat, bus) stopping at the open air museum (loved it), and at the mid way point on the rope way where we ate the black eggs! Sadly Fuji was playing hide and seek.

The next day after breakfast we headed back to Odawara, and took the shinkansen to Kyoto, where we stayed 3 nights. I think there are ryokan options but I stayed in a fairly function hotel near the main station (I really didn't spend any time other than sleeping in the room, as there was so much to see).

I did a day trip to Nara, which was amazing, really interesting (just watch out for the deer!) On the way back, by train, we stopped at Gekkeikan Okura Sake Museum which was interesting, and then back on the train for a couple of stops to Fushimi Inari-taisha which was very pretty.

Another day in Kyoto I had pre-booked tickets for the inside tour of the Imperial Palace (you need to book up several weeks / months in advance) and found the tour very informative. Then took the bus to the Golden Palace, which was my favourite of the trip - so pretty. After that we went down by bus to Nijo Castle, very pleasant to walk around.

We also went to the Gion district for a wander around, and to Yasaka Jinja Shrine, Sanjusangendo Temple, and Nishiki Market - basically a street full of food options. Kyoto Station is also very vibrant, food options, grocery store, department stores, a roof garden, steps that light up, and a walkway across the top which gives good views of the city.

After Kyoto we went to Mt Koya, where we stayed in a monastery - no food issues here, everything is vegan. Similar to the ryokan, hot spring baths, food int eh room, etc. but less pricey. Walked the length of the town, the cemetery is fascinating, and the large pagoda was very pretty. I was a little 'templed out' by this point tho, I have to admit.

I sent my luggage Tokyo - Kyoto and just took an overnight bag to Hakkone, same in reverse for Mt. Koya. Very simple, the bell boy helped me fill in the slips, and cost, IIRC, around $14 each way per case. With at least one of the companies you can track your bag on their online site.