Hi Sam,
Yes a 7-Day pass is a great option for you. It will costs about the same as a return Tokyo - Kyoto so any additional travel you may do becomes free! (see a little cost overview below).
Can we use JR Pass to ride Nozomi Express?
The JR-Pass can not be used on the Nozomi train, instead you can ride the Hikari trains that are almost the same, except that they stop at a couple more places on the way. The difference in time between a Nozomi and Hikari between Kyoto Tokyo is only 19 minutes for instance.
Can we use JR Pass within the city for subway?
The JR-Pass does not cover the the Subway, however you can use it in Tokyo on Local JR-Lines/ JR-Buses such as the Yamanote line.
Does JR Pass work for Keikyu Airport Express between Shinagawa and Haneda Airport?
The Keikyu line is a privately owned railway and the JR-Pass is not valid on it, instead you can use the Yamanote-sen from Shinagawa to Hamamatsucho and transfer there for free to the Tokyo Monorail that rides you in style to Haneda Airport.
Ticket Prices
Narita --> Tokyo ¥ 2,940
Tokyo --> Nara ¥ 13,750
Nara --> Kyoto ¥ 690
Kyoto --> Haneda (¥ 13,220 to Tokyo Station, ¥ 620 from there to Haneda Airport).
For a total of ¥ 31220 the 7 day Railpass is ¥ 28.300 saving you at least ¥ 2.920 + any local travel you may do on JR-lines in Tokyo.
Hope this helps :)
-Daniel