Hi, Giora,
I also have been to the Inuyama castle twice.
Near Gifu JR station, there is Gifu City Tower building that has a free observation area.
Near Gifu castle, Gifu city museum and Gifu Daibutu temple(big budda), Soho-ji temple is there.
I have been to both and they are not so impressive.
Gifu river is famous for Ukai, fishing with Cormorant. The season is over this year. In summer, every night, tourists boats depart.
The museum is new and opposite side of the Gifu river.
You can go to Kegon-ji by buses from Tanigumi guchi station of Tarumi railway or Ibi station of Yoro railway.
Both trains depart from Ogaki station.
The buses does not run fluently. Check time table in advance, if you use.
I can not find any interesting places along with the railways.
Sekigahara battle field is now local town with rice field.
If you like criming, Mt. Matsuo might be good. It was base of Kobayaka Hideaki.
His decision to attack the West determined the battle.
Fuwano seki gate is in Nara era not Heian. Sorry for my misunderstanding.
It is famous of the Jinsin civil war battle field in 672 AD.
After death of Tenchi emperor, his son and his brother had successor war.
The brother won and became Tenmu emperor. The son, prince did suicide.
Tenmu emperor made three gates to control traffic. One of them is Fuwano seki.
A small museum of Fuwano seki is near the gate locaton. Gate no longer remains.
As I wrote, Sekigahara is very important place in Japan history.
Main roads such as Tokaido, Isekaido, Hokkokukaido pass this area, because it is only place to cross Suzuka and Ibuki mountains. Then, Shinkansen and JR lines were constructed here.
Also, army corps gathered using the above roads and the plain became battle fields.
When I use Shinkansen trains, I usually see the Sekigahara battle field from the window.