7-day or 14-day Pass Needed?

Hello! Thank you in advance for your assistance, my itinerary is below, along with a few questions:

May 19-22: Tokyo (arriving in Narita); we will day trip to Nikko or Kamakura on May 21
May 22-23: 1 night in Gora
May 23-27: Kyoto; we will day trip to Nara on May 26
May 27-28 1 night in Miyajima
May 28-30: Tokyo again

Questions:
(1) It seems to me that a JR pass is needed, but do we need a 7-day or a 14-day? If a 7-day, when would we activate it?

(2) Access to and around Hakone/Gora: (a) How do we buy tickets for the Hakone railway if we are not buying the Hakone free pass? We are only spending one night at a Ryokan in Gora - we will probably explore the area around our Ryokan a little, but we are not necessarily planning on doing the whole Hakone loop.

(a) We are two couples travelling together, and someone suggested getting off the train at Odawara and taking a taxi to our Ryokan (Gora Hanaougi) - is it easy to find a tax that could fit 4 people plus luggage? How much would the taxi cost?