If I am at my credit limit and I pay only the minimum - can they add the finance charge and add on an additional fee. It seems that the finance charge is now increasing my credit limit yet I did not charge anything. Is this right, can the card company do that???

Yes. Welcome to the world of credit card "gotchas." It can be very expensive to max out your credit limit as it sets you up for extra fees should the finance charge push the balance over the credit limit.