Train Seat Fee: mandatory or optional?

Dear angel answer...

Checking fares and timetable at Hyperdia I got pretty confused about fares and seat fees.

I've checked trip from Tokyo to Nagano and the total is $8,200 (Fare $4,000 + Seat Fee $4,200), is the seat fee mandatory or can I travel with no seat reserved (in the cars with no seat reserved)? How does this seat fee works? At shinkansen trains is it possible to travel just paying fare, not paying for the seat fee? At my example seat fee charge doubles fare price!

Thank you in advance!

is the seat fee mandatory

Yes, it is.

or can I travel with no seat reserved (in the cars with no seat reserved)?

If you travel by non-reserved seat, you also have to pay the supplement which is usually around 500 yen cheaper than the one for reserved seats.

The underlying problem is the naming "seat fee" by Hyperdia that suggests that you only need to pay it if you use a seat. That is a misleading naming and is never used elsewhere. It is usually called the "limited express fee" or "supplement fee". And it is mandatory.

How does this seat fee works?

The basics are explained here, where the "seat fee" is referred to as "limited express fee":
http://www.japan-guide.com/e/e2359_001.html

At shinkansen trains is it possible to travel just paying fare, not paying for the seat fee?

No. If you don't want to pay the "seat fee", you have to use conventional commuter trains (but then you're probably better off taking a bus instead).