I've only used the Northern Kyushu 3-day pass once, and that was to take day trips out of Fukuoka.
Q1 - You can buy the pass at Hakata Station in Fukuoka for the advertised price. You just show your passport and they put a green ticket in a folder.
Q2 - Most of northern Kyushu can be reached from Hakata on day trips, such as
Karatsu (castle and pottery),
Imari & Arita (pottery),
Okawachiyama (small pottery town south of Imari),
Yoshinogari Koen (reconstructed Yayoi town in Saga),
Nanzoin (big bronze Buddha and temple in Sasaguri)
Hita, Mameda (old silver mining town and hot springs and Onta pottery)
If you want to take in sights around Nagasaki or Beppu/Yufuin, it might be more relaxing to stay in one of those 2 places for a night as they're reached by a longer train ride.
N. Kyushu doesn't lend itself for a complete circle route. If you go from Fukuoka to Nagasaki to Beppu, you have to backtrack almost back to Fukuoka.
The northern pass also gets you to Kumamoto City by Shinkansen if you want to check out the earthquake damaged castle and see all the landslide scars on the slopes around Mt. Aso. The rail line to Mt. Aso is still out, but JR has a special bus to get you around the damage.
Kokura is also accessible by Shinkansen on that pass if you want to see the Toto toilet history display there.
You could use up the 3-day pass on just day trips from Fukuoka unless you want to stay on either the east or west end of the island, or use other transportation to get away from the rail lines which would take time also (Kurokawa, Yamanami Highway).
Pass details:
https://www.jrkyushu.co.jp/english/railpass/railpass.jsp