What logical operators are used when more than one condition is set - are the conditions set in each tab AND or OR in relation to the previous condition tab?
That's difficult to answer. They're 5 independent conditionals. You should not have 5 conditionals controlling the same thing as that implies bad setup. If I had to give it an operator I'd say it's "OR", but that's not entirely accurate as again they're independent and unaware of one another.
Is it possible to use CB_Conditional to hide empty fields in the profile view for this situation?
You could set CB to hide empty fields within CB > Configuration > User Profile. Your conditional however shouldn't fail on profile view unless you've profile view disabled for your conditional.