The full name (and really most of the profile details) are stored in SharePoint, and can be viewed along with the rest of the profile fields in My Settings. The membership database is generally only used to store the username, password and email address
(the email address is duplicated in the SharePoint profile as well).
The full name is created from the First Name, Last Name entered when registering a user. It can be edited from the FBA User Management page as well as My Settings.
As for editing information on the My Settings page, I can edit it with a standalone SharePoint Foundation installation, however I don't have access to edit it on a SharePoint Enterprise installation with AD user profile syncronization. I'm not sure
what disables the editing, but my guess is that it has to do with the AD User Profile Syncronization settings, and the user is blocked from editing if AD Sync is setup.
As for allowing the creating user additional permissions - I don't think you'll need an additional field, the "Created By" field in the user information list should hold the user name. You will have to decide where to put the logic to allow editing based
on that field though. I can't think of a way you could edit the My Settings page to check for that information. You could change the FBA User Management page to check for that. Or create your own edit user page.