Reservations are separate from the app so any reservations that you have made you either need to show on paper or on your phone. If you haven't printed or downloaded any possible e-reservations you need an internet connection to show them.

You can add train to your journey when the app is offline but the app has to be online at least every 72 hours. Otherwise you can't create the QR code that is your ticket. 

https://www.interrail.eu/en/support/already-bought-pass/using-the-rail-planner-app/do-i-need-to-be-online-to-use-the-app-s-other-features-

The app needs internet:

• To activate the pass.
• To activate a new travel day. This also seems to be the case for continuous passes.
• To create a QR code (there's one for each travel day). I think this is done immediately when you add the first journey of the day to your pass. In any case, I made it a habit to display the QR code immediately after adding the first train of the day, just to be sure; after that, you don't need internet anymore on that day.
• At east once every 72 hours (if the previous points don't already do that).

Free-and dependable-wifi is available on many trains and also in many stations, but this varies per country. Swiss is the main exception in this-plus most in East-Europe. In Germany often labelled as WLAN. About all better HTLs and also about any hoStel will also offer it.

Switzerland is not really an exception. All main stations provide good free wifi.
Also a few trains have wifi - the Giruno and Astoro trains, but overall not too many… lets say 10% of the long-distance trains max.