Limited Express from Tokyo to Shuzenji

I will be staying in Izu for a few days in April and I am still a bit confused on limited express tickets and so on.

I want to travel from Tokyo Station to Shuzenji using the Odoriko. This way I don't have to transfer on the way and just take the bus to Shuzenji Onsen.

I understand that you have to pay a limited express ticket (with seat reservation, thankfully) for the Odoriko, but that you can pay the base fare with the Suica.

- I will probably arrive "logged in" with the Suica from another station in Tokyo, buying a limited express ticket on Tokyo Station. Do I just get on the limited express train and show my limited express ticket to the conductor on the way or are there other ticket gates to pass through, like when using the shinkansen?
- When getting off in Shuzenji, do I simply pass the ticket gates with the Suica?
- Or is it advisable to "logg out" in Tokyo Station and reenter the station with a paper ticket for the Odoriko instead?

Do I just get on the limited express train and show my limited express ticket
to the conductor on the way

Yes. You must have the limited express "paper ticket" or the express ticket online
by Ekinet, but you don't need to show it at the ticket gate. Just use your SUICA
card at the gate.

Ekinet membership
https://www.eki-net.com/en/jreast-train-reservation/member/wb/Login/Lo...

From April 1st to june 30th Ekinet provides small discount when buying through
Ekinet ticketless service.

https://www.eki-net.com/top/ticketless/

Further JR East desided to give up ticket checking by the conductor: The conductor
knows which seats are sold and are not sold and if she/he find there are passengers
on the "unsold" seats, she/he will ask them. If you have the ticket and sit in the
right seat, the conductor will just pass by.

https://response.jp/article/2016/04/19/273845.html

or are there other ticket gates to pass through, like when using the shinkansen?

No. As you say Shinkansen has its own ticket gate, but "ordinary" express does not.

- When getting off in Shuzenji, do I simply pass the ticket gates with the Suica?

Yes exactly. Have a nice trip!

Thank you a lot, that has answered all my questions ^_^.

Last time I had boarded a Shinkansen from Tokyo to Kyoto with the Suica still "checked in" in Tokyo, which of course resulted in error when trying to leave in Kyoto, so I wanted to make sure, how to best proceed in this case. (I understand I should have swiped the Suica at the Shinkansen gate as well...This time I am going to reserve Shinkansen online and connect this with the Suica so there should be no problems.)

I will also look into Ekinet. For some reason I cannot seem to add Shuzenji as arrival station on the english page. It works in the japanese version however. If I can't make it work, I will use the paper ticket, as described :-).

Shuzenji belongs to Izu hakone railway sunzu line, izu city,shizuoka prefecture.you can buy 1 day unlimited rail pass for 1040 yen,valid shuzenji station.Tokai bus 2 day pass=3900 yen, 3 day unlimited ride=4600 yen

Thank you, I checked these out but none will fit. The Izu Dream Pass would cover my route but I am staying 1 day to long for it to pay off. Guess I will have to pay step by step...
:-)