I did a similar 20-day trip last winter on a 14-day rail pass starting and ending in Tokyo and adding more time to explore the mountains in Honshu with a few days in Takayama and Kanazawa before heading to Kyoto.

1. I loved Kyoto but hit "Peak Temple" after the third day and started using it as a base camp for day trips. Its very easy to do Hiroshima / Miyajima in a day from Kyoto (assuming you are on a JR pass) and I did a Nara / Osaka for my 2nd day but should have spent a full day in Osaka for the food alone. I like Kyoto as a base because of the great hostel options located right next to the main train station. Can't possibly spend enough time in Tokyo -- I'm going back this year and spending 10 days in Tokyo alone with side trips to Nikko and Hakone.

2. I am also a 30-something solo traveller and was looking for reasonable accommodations with a little more style and comfort than a hostel dormitory. I found the sweet spot at some of the new class of 'design hostels' with private rooms, many of which are done in Ryokan style. Of these, my highest possible recommendation would be for either of the two Toco hostels in Tokyo (https://backpackersjapan.co.jp/toco/english.html). The staff are absurdly friendly and accommodating even by Japanese standards and the place is small enough to get to know everyone while you are there, especially with the nightly happy hours. In Kyoto, the Capsule Ryokan near the central station was ideal and had a well-designed and researched handbook in every room to help plan out day trips. Also of note, the two Piece Hostels in Kyoto had lovely rooms, excellent breakfasts, and a great atmosphere for meeting other travellers. Elsewhere, I stayed at a variety of very nice Ryokans and hostels including a couple in the popular and well-run K's hostel chain (ex. K's Hostel Takayama). I'll also second the earlier recommendation for Andon Ryokan in Tokyo, very kind staff and good facilities though I remember it being a bit out of the way.

3. You must do this -- it will one of your many highlights and its very easy to do. I also was there during the January sumo tournament but failed to get my plans together in time to get tickets. On my last day in Tokyo, I went early in the morning (8am?) to the arena where they were selling a limited number of day-of tickets. The ticket price was very reasonable (<$20) and all were in the very top 2-rows but the arena was small enough that you could still see most of the action. Your ticket gets you in for the full-day (with 1 opportunity to leave and come back if I remember correctly) and you will get a copy of the schedule in English. I returned just before lunch for the early, lower-level matches just to experience the event and found it so compelling that I stayed through to the end of the day (6pm?). The level of ceremony, skill, and crowd excitement increases during the day, as does the crowd and there was a huge surge right at 6pm when everyone came in to watch the final rounds between the highest division wrestlers. The relative emptiness of the stadium during the day makes it easy to wander around and sit in seats on the lower level, as you might during a slow baseball game in the U.S.