Hi Pim,
Let's get right in, there's a lot to be said and also have to correct the dates in my previous post, sorry for that (see question 4)!
1.) The JR Group is made consists out of 6 national groups nation wide and all of these accept the JR Pass, there are some rail lines in Japan that are held privately and don't accept the JR Pass. Think about metro networks but also local train companies in some places. In your itinerary you will only use of for the last part of travel to Mount Koya. and possibly to Mount Fuji. We have a very useful map which you can use to look up lines covered by the JR Pass. When using Hyperdia, unclick the box with Private Railway and NOZOMI / MIZUHO, this way you will find trains that are all covered by the pass, here's a somewhat more in depth guide.
2.) You can also use the SUICA card for travel to Kamakura as well as for travel to Nikko on both the JR and Tobu line.
3.) This is something I am actually not sure about. IC (such as Suica) cards used to be only valid in a certain region, such as SUICA in the JR East region (Mainly greater Tokyo). In 2013 it was made possible to use IC cards nationwide. However I am not sure if you can return one to say JR West if you bought one in JR East. Do note that the Suica card from the Suica + NE'X package is special version and only available for foreign visitors who land on Narita Airport - it's a great souvenir (or you could probably sell it on Ebay for a profit if you really want to get the deposit back).
4.) Sorry! you are absolutely right here! I checked your travel dates yesterday and counted 22 and had even write about how to best plan around it using the JR Pass. Then I double checked using the windows calculator function - like this - and it said 21. It does change things a lot! As the best situation would still be to use a 21 day JR Pass, however some re-scheduling would be needed. For instance, would it be possible to take a late noon train back to Osaka on the April 19th? This would allow you to cover all expansive travel parts and save you an otherwise ¥ 17,870 fare. An other option would be to make a day trip to Fuji from Tokyo move travel to Nagoya and subsequent days up by one day until the 10th where you otherwise may have visited Fuji.
5.) One of the most popular visits to Fuji is the Fuji Five Lakes area, of which the first lake is Kawaguchiko. Which is perfect for a day trip from Tokyo. A lot of travel to and from Fuji is via Tokyo in any event, so you may as well visit it from there - It would indeed be hard to reach Ise the same day.
HTML markdown does not work for safety reasons, however you can use the Daring Fireball markdown if you wish.
Hope this helps!
Daniel
PS: From time to time I find very funny alternative spellings to Japanese names on the forum. The best must have been when someone called Mount Fuji...Mount Fungi!