Hello Hannah,
I see you have a couple of questions, let's get started with them right away.
1.) Yes the JR Pass covers the Monorail from Haneda to Tennozu island.
2.) Yes, the Yamanote line is covered by the JR Pass.
3.) Going to Kawaguchiko is where things become a little more complicated. Using the JR Pass it is best to travel first to Otsuki station and take the Fujikyu line from there to Kawaguchiko. The JR Pass covers the train to Otsuki but you will need a normal ticket for the Fujikyu line (1,140yen). Here's the route from Shinagawa - Otsuki with the JR Pass.
4.) This is a pretty long route. Using the JR Pass, it's best to return to Tokyo and take the Shinkansen from there to Osaka. The full route can be found here. If possible, consider visiting Kawaguchiko as a day trip from Tokyo and travel to next day to Osaka, alternatively you could visit Hakone instead of Kawaguchiko, as it is directly on the route between Tokyo and Osaka.
5.) The JR Pass can be used on JR lines in and around Kyoto and Osaka. Other transport such as the subway is not covered. In Kyoto there's only limited service but you can see most of Osaka using the JR Pass. See our maps page for more details.
Hope this helps,
Daniel