Hi there!
A 14 day pass is ¥16800 more than a 7 day pass so you'll need to spend at least that on JR tickets for it to be worthwhile upgrading to a 14 day from 7. Lets get a general handle on the JR ticket prices for both of the options you list:
7 day pass costing
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Tokyo > Kyoto: ¥13,420 (164mins, 513km)
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Kyoto > Osaka: ¥540 (28mins, 42km)
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Osaka > Tokyo: ¥13,950 (194mins, 556km)
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Tokyo > Narita airport terminal 1: ¥3,140 (58mins, 79km)
JR ticket cost: ¥31,050 for 4 legs
As you can see you'll make a small saving with a 7 day pass here, but will retain travel flexiblity. You can also save money by using the JR Pass around Tokyo. For Tokyo JR travel it depends on your itinerary but with some planning you can save around ¥500-¥1000 per day.
14 day pass costing
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Tokyo > Maihama: ¥210 (15mins, 12km)
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Maihama > Tokyo: ¥210 (16mins, 12km)
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Tokyo > Nikko: ¥5,630 (121mins, 150km)
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Nikko > Tokyo: ¥5,630 (117mins, 150km)
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Tokyo > Odawara: ¥3,840 (35mins, 83km)
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Odawara > Tokyo: ¥1,450 (36mins, 83km)
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Tokyo > Kyoto: ¥13,420 (164mins, 513km)
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Kyoto > Osaka: ¥540 (28mins, 42km)
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Osaka > Tokyo: ¥13,950 (194mins, 556km)
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Tokyo > Narita airport terminal 1: ¥3,140 (58mins, 79km)
JR ticket cost: ¥48,020 for 10 legs
Note that I've only included the trip to Maihama (disneyland) here for your Tokyo travel. As I mention above you can usually add and extra ¥500-¥1000 per day savings with the use of the JR Pass in the city. Also, for Hakone the JR Pass will get you as far as Odawara. From Odawara you'll need to use the non-JR Odakyu lines up to Hakone itself.
Recommendation
As you can see, based on the two options you'll actually save more money by choosing a 14 day pass, especially if you take into account further travel within Tokyo. The difference between the Nikko Excursion Ticket and JR tickets is around ¥3500, so we can reduce your 14 day savings by this. I'm not aware of a Hakone 1 day pass (Odakyu do a 2 day or 3 day pass). If you let me know more about this I'll be able to cost it in for you.
Overall, I think you'll make some savings with both the 7 or 14 day pass (flexible travel around Tokyo), but they are unlikely to be large unless you do further trips, such as a return to Hiroshima or Kinosaki onsen while you are in Osaka. In the end, the main benefits of both 7 or 14 day pass in your case will be the travel flexibility to do these sorts of extra trips I think as any further travel you do over your listed itinerary will effectively be free.
As you'll be in Japan for 15 days, I recommend you purchase the [Narita Express + SUICA card deal on arrival at Narita]. This will get you into Tokyo on the first day and give you a free ¥1500 value for travel on JR and the metro. If you combine this with a 14 day JR Pass coverage you'll be able to use the Metro where needed (around 8 rides for ¥1500) and use the JR Pass for your travel on Tokyo JR lines for the majority of your travel.
Hope this helps!