Registration/Login: Are the following native to GForms?

1) A registration form where a user can create a profile

Yes, this is built int.

2) A login form for a registered user

Logins are handled by native WordPress functionality, not Gravity Forms. You are not collecting data and creating entries from a form submission; you're just logging in.

3) That (somehow) that a parent/child tree be password protected and one login permit access to all.

Not sure what this means, but it is no part of Gravity Forms, as Gravity Forms does not handle the login.

4) That every time the user sends an email via a GForm that the user's profile data be attached to the email.

I'm not sure what sort of form you are talking about, to send email, but you can modify the notification email to include whatever information you like, including information which is stored in the user profile.

Are the above function native to Gravity Forms, or easily done, or pre-documented?

See above. Some of it is not built in to Gravity Forms.

5) As a side wish, knowing that I'd probably have to custom code this, is it possible to create a history of the logged-in user's form submissions within the profile?

You would have to custom code this. You can pull out all the entries by user ID if you need to. I have not seen it done, but it's possible.

Your situation sounds very unique, and it will require a significant amount of work to make Gravity Forms do all this.