Several questions about management right

Hello Camille, welcome here! This is quite a long list of questions, our search bar up there would be very helpful to make it easier for you and other members so no one is overwhelmed trying to help out :wink:

That said, let me see how best I can help.

Sure, see: How to use category security settings to create private categories. If you want some extra customisations to list out topics that are for registered members only, see: Category Previews

They are shown a modal/message to sign up/register but I wouldn’t term that a denial

Yes, see: Sending Bulk User Invites

Yes, go to Admin > Customize > Email and select the email type you want to edit in the drop down

Yes, see the default invitee trust level site setting. This can also be set if you are using an external authentication or SSO provider for registrations and logins.

Off the top of my head, I do not think this is possible without a custom plugin, and if you want to go this route, spam will be a big issue and moderation will become a lot harder for large sites. You should/may still be able moderate, but I haven’t seen this request before, so I cannot speak to it and what would happen afterwards.

I am not sure I understand what you mean by Topics, but what you described can be achieved using Tags. See: It’s Time We Talked About Tags and Tags: category restrictions, tag groups, relationships

This is answered in the first topic I linked to.