The calendar new event page which you say should have a "Done" only has cancel or add buttons. Press either and what you have written disappears. No way to save calendar event that I can see
The "Add" button is the one you want.
Your phone likely has multiple calendars. It seems like you are currently adding new events to a calendar that is not visible.
First open the calendars app and note the word "calendars" on the bottom center (directly above the home button). Tap the word to configure which calendars are "visible". The calendar that you are currently adding events to is not visible. You can make all calendars visible simultaneously and you should find the events that "disappeared" as you created them.
The second setting to be aware of is "Settings > Mail, Contacts, Calendars > default calendar" (currently the last entry on that screen). This determines which calendar new events are added.