Hi there!
Lets firstly take a look at your costs to see if you could do your trip with a 7day pass + tickets:
- Tokyo > Osaka: ¥13750, (194mins, 556km)
- Osaka > Hiroshima: ¥9950, (104mins, 345km)
- Hiroshima > Osaka: ¥10520, (109mins, 345km)
- Osaka > Kyoto: ¥540, (29mins, 42km)
- Kyoto > Osaka: ¥540, (28mins, 42km)
- Osaka > Nara: ¥780, (59mins, 48km)
- Nara > Osaka: ¥780, (55mins, 48km)
- Osaka > Odawara: ¥12270, (159mins, 472km)
- Odawara > Ikebukuro: ¥1620, (67mins, 92km)
- Ikebukuro > Maihama: ¥380, (55mins, 25km)
- Maihama > Kamakura: ¥1050, (96mins, 66km)
- Kamakura > Tokyo: ¥890, (58mins, 53km)
- Tokyo > Karuizawa: ¥5750, (72mins, 146km)
- Karuizawa > Tokyo: ¥5750, (73mins, 146km)
TOTAL: ¥64570 for 14 legs
Please note that for Hakone I have used Odawara (from here you will need to use the non-JR Odakyu lines to get up to Hakone, I recommend the Hakone Free Pass for this. Also, for Disneyland/sea you should get to Maihama station and then use the Disney lines into the park proper. Aside from this, based on your itinerary the JR Pass will cover all the areas you mention - great! :)
As you can see you have costly travel in your itinerary all the way up until your Karuizawa trip, so I recommend that you opt for the 14 day pass, which will give you a saving of ¥19,470 over purchasing standard tickets. The only way a 7 day pass may make sense is if you were to rearrange your trip with the majority of the costly travel in the first 7 days, but I think you have a well spaced itinerary so recommend you stick with it.
When you are in Tokyo & Osaka you can get good use out of the JR Pass as both cities have a good internal JR network (map of JR lines in Tokyo). However Kyoto and Nara don't have internal JR and all will need you to use non-JR forms of transport at some point. It does depend on how much travel you want to do in these areas but I recommend you either purchase standard tickets/prepaid cards or a municipal pass over the Kansai Thru Pass as the JR Pass will cover your travel between these main locations and you only require internal travel.
So the short answer is: a 14 day rail pass and pay as you go internal subway/bus tickets as you need. :)
Hope this helps!