Can invite module work for existing users?
The Invite module serves to invite those who are not yet existing users to become ones. That's why you cannot use it to invite existing users.

Anyway, when using UR, at existing user's public profile, there's a Become USERNAME's friend link which you can use to befriend an existing user. This is an equivalent of the Facebook Add Friend button.

With the UR's user_relationship_invites module enabled, you can give users the option of specifying a relationship when inviting new users, e.g. friend. Together with the Contact Importer this would be an equivalent of the Facebook Find friends page.