Itinerary Review

Me and a friend are going to Japan for 2 weeks in July and I just wanted to see if the things in my itinerary are do-able, and our main city of stay will be Tokyo:
Ok so we'll be arriving in Tokyo on July 15th and spend the 16th just exploring Tokyo.
From the 17th-19th We'll be visiting Osaka and seeing the sights and head back to Tokyo by late day of the 19th .
On the 20th-21st we'll be heading from Tokyo to Mt Fuji by bus from Shinjuku station and head back to Tokyo by the 21st
Then on the 22nd-25th we'll be visiting Kagoshima/Tanegashima and head back to Tokyo by the 25th, and for the rest of our stay from the 26th-28th we'll be in Tokyo.
I just want to see if this itinerary is do-able or not and if anyone has any suggestions.

Also to get to all these destinations we'll have a JR pass but I just want to know is it worth it to get a Green over an Ordinary 2 week JR pass and are there any major differences between them?

And for the bus from Shinjuku to Mt Fuji, do we have to reserve a seat on the bus before hand or just buy the bus ticket at the bus station?

Finally, are the trains that run from Hakata Station to Shin-Yatsushiro and then from Shin-Yatsushiro to Kagoshima-Chuo Station covered by the JR pass?

It is possible, but could be put into much more efficient order, for example:

Tokyo - Mount Fuji - Osaka - Kagoshima - Tokyo

Also to get to all these destinations we'll have a JR pass but I just want to know is it worth it to get a Green over an Ordinary 2 week JR pass and are there any major differences between them?

First of all, note that the distance between Osaka and Kagoshima and even more so the distance between Kagoshima and Tokyo is very large. It takes almost ten hours from Kagoshima to Tokyo by train! I recommend to consider flying from Osaka to Kagoshima and from Kagoshima to Tokyo instead. Here is some information on saving on flights:
http://www.japan-guide.com/e/e2364.html

And for the bus from Shinjuku to Mt Fuji, do we have to reserve a seat on the bus before hand or just buy the bus ticket at the bus station?

You can make the reservation at the Shinjuku bus terminal just before boarding, assuming that the bus is not booked out. If it is a very bus day, you might want to make advance reservations.

Finally, are the trains that run from Hakata Station to Shin-Yatsushiro and then from Shin-Yatsushiro to Kagoshima-Chuo Station covered by the JR pass?

Yes!

Don't bother getting the green pass, the ordinary carriages on the shinkansen are very comfortable with lots of leg room.

You dont have to and should not travel back to Tokyo each time. It's a waste of effort, time and money.