7-day pass

Hello,

We're trying to figure the most effective way to use JR Pass during our Japan trip from May 3 to May 15.

Below is our schedule.
1. Tokyo: Stay in Tokyo from May 3 to May 7
2. Kyoto: leave Tokyo on May 7 to go to Kyoto and stay here until May 11.
3. Osaka: Leave Kyoto on May 11 to go to Osaka and stay here until May 14.
4. Back to Narita hotel: Leave Osaka on May 14 and stay in Narita hotel that night.

We do not need JR pass for the trip from Narita airport to Tokyo because our tour guide will help us with that.
We will need to use trains to go from Tokyo to Kyoto, then from Kyoto to Osaka, and finally from Osaka to Narita hotel. If possible, we also like to use JR pass within Tokyo, Kyoto and Osaka.

We have few questions:
1. Which pass should we buy? will the 7-day pass work for us?
What is the timing for using the 7 day pass? Does it count based on the time that we start to use the pass? For example, if we start using the pass at 7pm on May 7, is it still valid to use it before 7 pm on May 14?
2. Do we still have enough time to order the JR passes online and pick them up at the train station?
3. If the 7 day pass works for our plan, which trains we should take from Kyoto to Osaka and from Osaka to Narita?
4. Can we use JR Pass within Tokyo, Kyoto and Osaka. Our tour guide said that JR pass doesn't help much within a city . Is it true?
5. If the 7-day pass doesn't work the above plan, Is there any regional pass that works for us?

Thank you,
Ly