Tokyo-Kyoto-Osaka

Good afternoon, we are planning our first trip to Japan and are landing at Haneda on March 28th 2026. This is a 13 day trip between Tokyo-Kyoto-Osaka. I am beginning the research for our group (6-7 people) and have questions regarding the JRPass and if this is the best option. Below is the itinerary followed by some questions:

March 28th - Land at Haneda, train to Tokyo
March 29th - April 3rd - Explore Tokyo
April 3rd - Afternoon train to Mount Fuji, spend the night
April 4th - Midday train to Kyoto
April 4th - 6th - Explore Kyoto
April 7th - Morning train to Osaka
April 7th - 10th - Explore Osaka
April 11th - Train from Osaka to Haneda for Departure

Questions:

Would this itinerary constitute the purchase of the JRPass?
Would a regional pass be more logical? (The arch option doesn't seem feasible)
Does this pass include local trains around each of these cities? Buses? Etc?
We plan to have day trips outside of these cities, does this pass work for this as well?
Is there any other options that would suite this itinerary better?

I'm sure I am missing some things but these are my initial concerns. Any help with this will be greatly appreciated!