railpass 14day's?

Hello,

We're going to Japan in July. We're 4 persons ( 2adults and 2 children 14 and 15 years old).
the plan is the following:

• day 1: arriving in Osaka -day2: leaving for Kyoto for 4 nights+ daytrip to Nara from Kyoto
• day 2 kyoto -day 3: daytrip Nara -day 4: kyoto -day5: leaving for kyosan -day 6: trip to hiroshima
• day 7: hiroshima
• day 8: naoshima -day 9: kanazawa
• day 10: kanazawa -day 11:shirawaka-go -day 12: takayama -day 13: takayama
• day 14: mount fuji
• day 15: mount fuji
• day 16: mount fuji -day 17: tokyo
• day 18;trip to Nikko -day 19: tokyo -day 20: back home

Is it best to take a JPrailpass starting from day 4 for until day 17 , 14 days . Or do you have any other suggestions.

I have an other question: If you want to take the Nozomi train, do you have to pay a supplement or buy a new ticket?

Thanks in advance

Sabine