Hi Chloe,
The key thing to remember is that the difference in price between a 7 and 14 day Japan Rail Pass is ¥16,800. As a 3 day Hokkaido pass is ¥15,000, I think I recommend spending the extra ¥1,800 for the 14 day pass and retaining some flexibility in your plans if you are just choosing between the 2 options. Similarly, you'll need to spend at least ¥16,800 on additional JR tickets to make a 14 day pass worthwhile over a 7 day pass + tickets.
Looking at your plans you have 5 days of confirmed travel, with one additional trip to Kumamoto or Noboribetsu possible within your 7 day validity period depending on your plans. As the Fukuoka/Hakata to Kumamoto return is around ¥10,000 and Sapporo to Noboribetsu return around ¥8500, I recommend using the 7 day JR Pass validity period for the Kumamoto trip.
As to whether a 7 day pass + tickets or 14 day pass is the best option, you'll need to spend at least ¥16,800 on separate JR tickets in addtiion to the 7 day pass for the 14 day pass to be worthwhile. Your trip to Noboribetsu will likely come to around ¥8,500 return so this means you'd need to spend a further ¥8300 on JR travel to make a 14 day pass worthwhile. You mention further sightseeing within Sapporo, and although you'll be using the non-JR metro within Fukuoka, you can do easy day trips by JR to Hiroshima/Miyajima, Nagasaki, Dazaufu-Tenmangu or even Yufuin/Beppu. Whether you hit ¥8300 extra JR travel is down to your plans so I think I'd base my decision on whether I though I was likely to do that much more travel at either end of my trip or wanted to retain the flexibility to decide when I was there.
The exchange office in Fukuoka is at the main station, called "Hakata" station for historical reasons. The JR-Kyushu ticket office can exchange your JR Pass from 5:30-23:00, and there is also a travel service centre open from 10:00-20:00 for exchanging JR Passes and further queries.
Hope this helps!