Hi,
Well first - April has only 30 days in it.
A 7 day pass can pay off from your itinerary. It will only help you part way to go see Hakone though. You can get a Hakone Free Pass and see a lot of different places there though. This info can help you:
http://www.japan-guide.com/e/e5200.html
Otherwise, you can go all over using the JR Pass.
Since you are already seeing Tokyo, you can largely skip Osaka during the day. After the temples of Kyoto close down around 5PM, you can zip over to Osaka for the evening. At night the city comes alive and has some great places to see, such as Dotonbori, plus the night views from the Umeda Sky Bldg, and Abeno Harukas Bldg are stunning.
There are many nearby sights for you to enjoy, including Nara, and Himeji.
Just one snag if you are entering and leaving Japan through Narita Airport in Tokyo. Since you will be in Japan 8 days, if you get the JR Pass you can use it for the Narita Express train to go into Tokyo from Narita, or on the 3rd to go to Narita, but not both. A day used for a 7 day pass starts and ends at midnight, so if you arrive at 6 PM in Narita airport and activate your pass, your first "day" is just 6 hours. Fortunately you can go cheaply by other means - look into taking a cheap 1000 yen bus.
If you wish to stay overnight in Osaka on May 2nd, it is possible to go from there to Narita airport for your flight, but you'd have to leave pretty early in the morning for it. You might consider returning to Tokyo in the evening of May 2nd to make it easier for you.
Good luck!