Note: If it's a free event, simply let users create account from the account creation page (https://youreventname.dryfta.com/en/attendees-authors-registration) and enable option to automatically assign "attendee" role to all users who create their accounts.
Go to Contacts Settings to enable Invite-only registrations. Under Enable Invite-only registrations, generate a code and share that with your prospects. They would need to enter that code when registering for the event. See screenshots.
That's all you need to do to set up your invite-only event registration form.
To customize the account creation form with your own questions including event dates on which they would be available, go to Forms & Data > Form builder and click on Add New Field button in the top right corner.