Hello Cecilia,
I see you have a couple of questions. Let's get right in!
Is there any other transportaition included in Tokyo with this pass?
Yes, the JR Network covering Tokyo and around is included in the JR Pass. This includes the JR Yamanote loop line, that makes a circle around Tokyo and stops at major stations like, Shinjuku, Ueno, Harajuku, Ikebukero and so on. We also have a map with all JR Lines here.
Is there a Train included with this pass going to Kamakura and Nikko from Tokyo ?
Yes, you can use the Shinkansen bullet trains for part of the way even. The full route is: Take a Shinkansen from Tokyo station to Utsunomiya, there transfer to a local line to Nikko on the JR Nikko line.
Can you actived your pass 2 day after you exchange or it has to be the same day ?
This is also possible. When you activate the JR Pass, you can select a first day of use, up to 30 days later than the day of exchange.
Hope this helps,
Daniel