1.Can Permissions be inherited from parent folders by default?
Yes, permission is inherited from parent folder.
Please note that shared icon of child folders will not appear on the side of the user who create share. Shared icon of child folders will appear only on the side of the user to whom the folder is shared.
2. How do you rename groups?
As of this writing, it is not possible to rename groups. If you need group with a different name, you need to create a new group.
3. If a folder has been renamed only that user can see the change, is this usual behaviour?
Yes, this behaviour is by design. Any name change of a shared folder once it is shared will not appear to other users who use that shared folder already. However, the other person who use the shared folder rename it as he wish. Again, it will not affect name of the original folder.
Please note that change in name of any child folders inside the shared folder will appear as such to the other person who use the shared folder.
4. The admin account can't see folders that aren't shared with the admin user, is there a way of making this happen? Can we set it so the admin user can see all folders regardless of share permissions?
No, this is not possible by design. Sharing is the only way to see other users' folders for all users including admin.