Hi there!
I love your plan, there's a lot of variety in it and you also have a good amount of time to spend in each place. Amanohashidate is a great visit as well, you could also hop over to Kinosaki-Onsen from there to enjoy a night at an Onsen Ryokan.
As for what JR Pass to use. I think that either the 14 or 21 day JR Pass would be a good option. The 14 day JR Pass could be used from April 14th - 27th and you would cover most of your otherwise expansive travel with it (assuming that you would be back in Tokyo by the end of the 27th). Travel around Tokyo is not very expansive and you could buy a [Narita + Suica package] for the first couple of days. This package gives you a discounted ticket on the Narita Express to Tokyo and also has a Suica card which makes travel around Tokyo very convenient.
If you really don't want to think about tickets, consider the 21 day JR Pass for your whole journey. It would basically cover all rail travel, with the exception of local travel in Hakone. The difference in price between the 21 and 14 day JR Pass is 12,600 yen. I don't think that you would do this amount of travel right in Tokyo. A return Tokyo - Narita Airport is roughly 6,000yen for normal tickets. However you could consider a 21 JR Pass is you want to the extra flexibility or in case that you would wish to make a day trip from Tokyo, to Nikko for instance.
I hope this helps!
Daniel
PS: for your travel in Hakone, take a look at the Hakone Free Pass it may help you get some good discounts, depending on what you wish to see in Hakone.