Hi there,
It's hard to say regarding the Green Pass. Certainly when I have used it in the past it has been very quiet and the seating is much bigger than that of standard class. Please note that all seats in Green Class must be reserved (normally you can simply reserve just before you travel, however all seats must be reserved).
As all seats in Green class must be reserved and you are planning on travelling during rush hour periods, I recommend that you purchase a standard class JR Pass and reserve your seats as soon as you arrive in Japan. Seat reservations are free for JR Pass holders and are very simple to make at the station - you can learn more how to make seat reservations here.
About your hyperdia search, I actually couldn't replicate your search, and the price of the train was very similar to Kobo(hyogo) and Shin-Kobe (they are very close, just Shin-Kobe is the bullet train terminal so I used this as it will be the faster option). Here is the search again using Kobe(hyogo):
-
Sendai airport > Sendai(miyagi/jr): ¥630, (27mins, 17km)
-
Sendai(miyagi/jr) > Rifu: ¥230, (16mins, 12km)
-
Rifu > Sendai(miyagi/jr): ¥230, (17mins, 12km)
-
Sendai(miyagi/jr) > Rifu: ¥230, (16mins, 12km)
-
Rifu > Sendai(miyagi/jr): ¥230, (17mins, 12km)
-
Sendai(miyagi/jr) > Kobe(hyogo): ¥22110, (356mins, 941km)
-
Kobe(hyogo) > Kyoto: ¥1050, (55mins, 75km)
-
Kyoto > Kobe(hyogo): ¥1050, (56mins, 75km)
-
Kobe(hyogo) > Shinjuku(jr): ¥14470, (252mins, 593km)
-
Shinjuku(jr) > Urawa: ¥380, (27mins, 21km)
-
Urawa > Shinjuku(jr): ¥380, (28mins, 21km)
-
Shinjuku(jr) > Narita airport terminal 1: ¥1450, (100mins, 86km)
TOTAL: ¥42440 for 12 legs
As you can see, using Kobe(hyogo) is slightly cheaper than the 14 day rail pass overall, however it involves slower trips to and from Kyoto so there is a tradeoff here. Overall they are much the same in terms of costs I think.
On balance as the difference in prices is so little I think I'd advice to go with the rail pass vs individual ticket so that you maintain flexibility to change plans/stations and not affect your overall travel budget, however this may depend on the degree to which you have your travel plans finalised.
The options you deselected on Hyperdia were correct, so I wonder where your original costs for the Sendai > Kobe came from! As a check I am sure the price you have of ¥11660 is not correct (maybe a system error?) as a single from Tokyo to Osaka (which is much less than the journey you are making here) is ¥13,950.
Hope this helps!