No, currently it is not possible. You can make the field either visible or in-visible. To hide/show fields, Goto Setup >Admin Settings > Field level security >select profile & module to hide/show fields For More refer Field Level Security.
Yes, you can make a particular Topic as private which is not visible under customer portal. Goto Solutions tab > right click on Topic and select the option Make all solutions private. For more refer Create Topics.
No, not necessary. Zoho Support creates a default Department as General. You can setup Mail Server settings for the default Department or create your own Department. Refer the below links for more: Mail Server Settings Department