Public profiles

What do you mean by "My profile shows that everyone should be public"?

What do you mean by "my relatives cannot open anything"?

Default on Geni is that only the Geni-user can edit the profile of themself.
If you want all your relatives to be able to edit your profile of yourself, you can change your setting to allow that.

See "Request to edit Profile" on your Profile - so am guessing that you have edited your settings to let family edit your profile of yourself (and since I am not in your Family Group, that Request to edit would amount to a Request to join your family group)

All your immediate family except your mother is Private. If they are living, you cannot change that.

If your relatives are not Geni members, they cannot edit anything on Geni.
You can invite relatives to join Geni by entering their email in their profile - if they accept the invite, then default will be only they can edit their own profile, and they will be able to see and edit the private profiles of your mutual relatives same as you can.