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 
That said, let me see how best I can help.
Sure, see: Understanding groups and category permissions (security settings). 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
message.
Yes, see: Send 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 Set up structured tagging with tag groups and category tag restrictions
This is answered in the first topic I linked to.