Users Posting under different categories

Sorry guys, im newbie, I am trying to look these quetions up with no luck:

I want users to be able to create topics and at the end choose according to which group they would like to add it.

1- Can I have it Uncategorized by default w/o the user getting an error message that they did not choose a group (and their whole message gets deleted)

2- How can I control the groups as an admin? can I view all groups and edit them?

3- it looks to me that users can only post on groups that they joined. This is not right, so where’s the option to change that

thanks in advance