Generally longer term trips mean a slower travel pace and that may make it harder to make a rail pass pay off. Without seeing your whole plan and how much time you plan to spend at each destination, it is really impossible to see what pass works or not. Although from your list so far I question if a 21 day pass would work for you.
Have you tried to use the rail fare calculator on
https://www.japan-guide.com/railpass/yet?
It's simply impossible to say whether going to Kanazawa and then starting your JR pass or flying to Fukuoka first is the way to go without looking at all the variables like what airfare you'd be paying, if you'd be using a low cost carrier like Jet Star or ANA's Experience Japan Fare etc, how'd you get to Kanazawa in the first place... you need to give more info. Obviously if you flew to Fukuoka you'd make using a JR Pass harder to justify - but it is up to your destination order and naturally the aim of going to Japan is to see a lot of nice places and not just to make a rail pass pay off.
The goal is to pack in as much long distance travel as you can into as short a time as possible (without overdoing it of course) to get the most value. It is not a question of JR covering "day trips" - it covers JR routes and it's up to you to use your time effectively. Of all the places you mention, JR will take you to most of them. For Hakone though you should look into using a Hakone Free Pass.
http://www.japan-guide.com/e/e2358_008.htmlThe Loop Course is a common way to see all the sights and takes a full day.
http://www.japan-guide.com/e/e5210.htmlFor Yanagawa it's better to use a Nishitetsu train (not covered by the JR Pass). The fastest is about 55 minutes from Tenjin Stn in Fukuoka and 850 yen one way.
A JR Pass can help you to a certain extent in the cities but the value it adds is limited and you generally should not rely on it alone to get around at your destinations. In Tokyo you can use it on the Yamanote Line and it goes to several popular tourist areas. Likewise in Osaka you can use it on the Loop Line. In Hiroshima you can use it on the Loop Bus and for the JR ferry to Miyajima. For Kyoto you can use it on the Nara Line to see the Fushimi Inari Shrine for example, as well as Nara of course. You can also use it to go to Arashiyama on Kyoto's west side.
But there are local transport passes that may be more helpful for you. In Tokyo there is a good 48 & 72 hour subway pass
http://www.tokyometro.jp/en/ticket/value/travel/index.html#anc03and in Osaka the Osaka Amazing Pass might be worth it for you if you plan to go to the places it features.
https://www.osp.osaka-info.jp/en/For Fukuoka, many good places are near a subway station, and there is a good 1 day pass and even better 2 day subway pass.
http://subway.city.fukuoka.lg.jp/eng/fare/two/
http://www.thejapanfaq.com/fukpic.htmlhttp://bit.ly/1KAm8cUThe IC cards are convenient in that you don't need to carry around a pocketful of change all the time, but aside from that there is negligible savings from them.
