Help with 18-day itinerary?

A 14 day pass would certainly pay off if you are going from Tokyo to Kyushu and back to Kansai. You can activate it on Day 5 as you mentioned. But you can't use the JR Pass all the way to Hakone, since there is no JR service there. You can take a bullet train from Mishima to Nagoya, then continue to Takayama.
The following info can help:

http://www.japan-guide.com/e/e5200.html

http://www.japan-guide.com/e/e2358_008.html

http://www.japan-guide.com/e/e5206.html

http://www.japan-guide.com/e/e5210.html

Unfortunately I sorely doubt you can fit in the Tateyama Kurobe Alpine and then do a lot of travel down to Kyoto - just too hectic. Your 2nd idea is better by skipping Osaka - since you are already seeing Tokyo, you can skip it during the daytime anyway. After the temples close in Kyoto around 5 PM, it is easy to zip over to Osaka to spend the evening. Dotonbori in Osaka is a must see place at night:

https://www.youtube.com/watch?v=COJ_MeXbeow

Seeing the city from the Umeda Sky Bldg and Abeno Harukas Bldg are also spectacular:

https://www.youtube.com/watch?v=xb1wuyAGgyQ

https://www.youtube.com/watch?v=-AyUZmGbW8g

Going to see the wisteria is very ambitious and I wouldn't recommend doing it unless you can overnight in Kitakyushu - hopefully your flight on Day 18 is not in the morning. First off, you are never going to get there looking for "Wisteria Lane".
It might not be worth the trouble if it's not in bloom, so you should call first.
The place you want to go to is Kawachi Fujien, and it is very time consuming to get to. You can take the bullet train to Kokura, switch trains to go to Yahata, then catch a taxi (pricey) or sneak on a shuttle bus going to a nearby hot spring place, Ajisai no Yu (1X/hour at 25 min after the hour, exc. 2PM). The garden is 400m uphill from the spring. It'd be very hard to squeeze in Rabbit Is. together in one day. But if you overnight in Kitakyushu, nearby Sarakurayama gives a splendid view over the city:

https://www.youtube.com/watch?v=Hl1pCW9Ibbk

Kokura Castle is also nice to visit, and going down to Hakata to see the Nanzoin Temple is also marvelous:

https://www.youtube.com/watch?v=lT4VQFmHFyM

Good luck.

Thank you for all the advice! Much appreciated!

I will try calling before I go to Kitakyushu but I will be staying overnight in Hiroshima the night before. So the journey there won't probably be too bad I hope. It's the travel back to Kyoto that's going to be long. You're right, I can't fit rabbit island.

Sarakurayama and Nanzoin temple looks wonderful but I won't be able to stay overnight in Kitakyushu. My flight the next day is at noon.

When I am in Tokyo, do I just purchase a Suica card?

Thanks!

A Suica card saves you from carrying a bunch of change, but there is no real savings in it. If you are doing some sightseeing in Tokyo, a 3 day subway pass is very economical, though. See:

http://www.tokyometro.jp/en/ticket/value/travel/index.html#anc03

A day trip to Nikko or Kamakura would also be well worth it. There are some regional passes you can use:

http://www.japan-guide.com/e/e2357.html

http://www.japan-guide.com/e/e3800.html

http://www.japan-guide.com/e/e2166.html