Hi Rob,
Here are some answers for you:
General Coverage
Here's a great map of the general routes you can take with the JR Pass. The JR pass allows free travel on all JR lines in Japan including all bullet train lines. About trains, the general rule of thumb is that you can use all daytime trains and bullet trains except the Nozomi and Mizuho class bullets (There are always alternative Hikari and Sakura class bullets you can take instead that travel on the same lines and reach the same top speeds but stop at more stations en-route).
Night trains
Normally sleeper/night trains are not included with unlimited travel rail passes, however with the Japan Rail Pass certain sleepers are covered: the Tokyo-Osaka Sunrise Seto & Sunrise Izumo, the Tokyo to Aomori Akebono and the Aomori to Sapporo Hamanasu). These sleepers provide very basic free accommodation to JR Pass holders (think carpeted areas with sheets and pillows). You'll need to pay extra for a proper berth or room if required. The main problem with the night trains is that they must be reserved in advance, and for JR Pass holders can only be reserved once you arrive in Japan and can present your rail pass. For those travellers with a 21 day pass this can be a good option, however for those on 7 day passes the sleepers may all be booked up by the time you arrive.
Planning your trip
If you're going for speed then the bullet trains will always be your best bet so try and go for these if possible. You can look up precise train timetables and costs at the Hyperdia travel planning website and learn how to use Hyperdia for JR Pass holders to plan your trip through our video blog.
Reservations
Seat reservations are 100% free for JR Pass holders, however can only be made in person at the JR ticket office once you have activated your rail pass in Japan. We have a good blog post on the reservation process here.
Your journey
Here is a sample itinerary for you valid for the Japan Rail Pass and based on your title. Note that for Hakone the JR Pass will take you as far as Odawara. From here you'll need to use the non-JR Odakyu lines to get you up to Hakone itself. I recommend the Hakone Free Pass which you can purchase on arrival at Odawara. I have also assumed you'll arrive and depart from Narita Airport:
-
Narita airport terminal 1 > Tokyo: ¥3,140 (62mins, 79km)
-
Tokyo > Kamakura: ¥2,100 (47mins, 51km)
-
Kamakura > Odawara: ¥740 (49mins, 41km)
-
Odawara > Kyoto: ¥11,950 (160mins, 429km)
-
Kyoto > Osaka: ¥540 (28mins, 42km)
-
Osaka > Hiroshima: ¥10,150 (104mins, 345km)
-
Hiroshima > Nagasaki(nagasaki): ¥12,850 (190mins, 434km)
-
Nagasaki(nagasaki) > Tokyo: ¥24,940 (499mins, 1328km)
-
Tokyo > Narita airport terminal 1: ¥3,140 (58mins, 79km)
JR ticket cost: ¥69,550 for 9 legs
Hope this helps!