Hi there!
Hard to say really. The trip from Tokyo to Odawara on bullet train is ¥3640, whereas the romance car will be around ¥2020 one way or an extra ¥1100 for the Hakone Free Pass from Shinjuku.
If you thought that the travel you list is the only travel you will do, then the costs of the tickets for the 7 day period from odawara onwards will be:
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Odawara > Osaka: ¥12,070 (190mins, 472km)
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Osaka > Kyoto: ¥540 (29mins, 42km)
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Kyoto > Tokyo: ¥13,420 (161mins, 513km)
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Tokyo > Narita airport terminal 1: ¥3,140 (58mins, 79km)
JR ticket cost: ¥29,170 for 4 legs
As you can see there is a very slight saving with a JR Pass doing it this way round.
In the end, I think your travel plans are a borderline case. If the travel you list is the only JR travel you intend doing, then there won't be much difference between individual tickets and a JR Pass. If, on the other hand, you wanted some travel flexibility (eg, possibility for day trips from Osaka to Himeji, Nara, Hiroshima, Kurashiki), then the JR Pass will give you this and not increase the overall price of your travel.
I can confirm that the Narita Express will allow you to travel from airport to Tokyo on day 1 and back to the airport on day 9 (Provided your airport is Narita airport and not Haneda airport).
Hope this helps!