Yes you always need to log the train on the app (whether there are required reservations or not).

The departure day will be counted as a travel day.

Does this count as a travelday?

Yes it does since your “interrail discount” is in fact you using your interrail that day.

When you board, the conductor will need to see your berth reservation, and then when they see it is based on interrail pricing they will want to see your interrail ticket.