Hey Flavius,
Unfortunately right now EasySocial does not have a closed invitation system as you can only have public / private registrations. You can however, manually create the user's account from the back end though.
As for signing up for profile type, I am not really sure if I get you here but when the user tries to register, he / she could choose which profile type they want to register with.
Alternatively, you can also disable specific profile types and only allow them to choose those that you want to allow.