JR pass to Nikko from Shinjuku?

Can you please help me to decide.

We want to go to Nikko from Shinjuku on December 13th. I'm not sure whether to activate our 7 day JR pass on Dec 13th in order to use it to go to Nikko without having to pay the fares, or activate it on Dec 14th. If we activate the JR pass on Dec 14th then I guess the last day to use the pass will be Dec 20th? We will be in Osaka on the 19th Dec from Hiroshima and I don't think we will be going further on Dec 20th other than walking around in Osaka. We will leave Japan from Osaka on Dec 21st.

What is the best way to travel from Shinjuku to Nikko using a JR pass? Do we have to pay any supplement on top, if so how much? I was thinking of getting the World Heritage Pass as it includes admission to the shrines. But since we'll have a JR pass we might as well make the most of it. Do you think we'll save a lot by using the JR pass?

What are the costs for admission to Toshogu Shrine, Futarasan Shrine and Rinnoji Temple, travel on buses in central Nikko and on trains between Nikko and the Kinugawa Onsen area? If it's not much of a saving doing by using the JR pass, then I will get the world Heritage pass.

Thanks

What is the best way to travel from Shinjuku to Nikko using a JR pass?

By shinkansen via Utsunomiya:
http://www.japan-guide.com/e/e3805.html

Do we have to pay any supplement on top, if so how much?

No, not if you stick to JR trains. Note that a supplement is required on the direct JR/Tobu limited express trains between Shinjuku and Nikko.

I was thinking of getting the World Heritage Pass as it includes admission to the shrines. But since we'll have a JR pass we might as well make the most of it. Do you think we'll save a lot by using the JR pass?

Yes, about 2500 yen. Instead of the World Heritage Pass you can buy the 1000 yen combination ticket for the shrines and temples of Nikko. A shinkansen ride between Hiroshima and Osaka would provide even bigger savings (about 10,000 yen).

What are the costs for admission to Toshogu Shrine, Futarasan Shrine and Rinnoji Temple, travel on buses in central Nikko and on trains between Nikko and the Kinugawa Onsen area?

It is all explained here:
http://www.japan-guide.com/e/e3800.html
http://www.japan-guide.com/e/e3805.html

Thank you again Uji,you have been a great help all the way.

Regarding the 1000 yen combination ticket for the shrines and temples of Nikko, does this include bus ride within Nikko and the ropeway to Lake Chuzenji and bus back to Nikko? If not, how much extra?

Thank you

Regarding the 1000 yen combination ticket for the shrines and temples of Nikko, does this include bus ride within Nikko

No. That just costs a few hundred yen separately.

and the ropeway to Lake Chuzenji and bus back to Nikko?

There is no ropeway to Lake Chuzenji. Access to Lake Chuzenji is also by bus. The cost is about 1000 yen one way and not included with the World Heritage Pass, either.

All the details can be found here:
http://www.japan-guide.com/e/e3800.html