Osaka-Tokyo-Osaka

Hi,

I am planning a short trip to Japan (oct 12-oct 16) and need to know if the JR 7 day unlimited pass can cover all my travel and if it is worth it?
I am flying into Kansai International airport in Osaka- can i pick up my JR pass at the airport and take the train to my hotel in Osaka from there? What train should i look for?
The next day, i plan to take the bullet train to tokyo and spend 2 days there.
On the 3rd day, i plan to take the bullet train back to osaka and spend my last day there. can i travel around Osaka with the same jr pass?

Can a 7 day unlimited JR pass cover my whole trip?