Yes, you can remove the public signup page and add users through invites.
As soon as you click the Use the template button, you'll be immediately taken to Softr studio and can start working on the template. If you don't have a Softr account, you'll be prompted to register first (a Free account will work just fine).
All the Softr templates are completely free of charge and can be used with any of the Softr subscription plans. However, Free and Starter plans have a limitation on the number of application users, so you might need to consider a higher subscription plan if you need more people to have access.