Yes, you must have a barcode to sell cards. This is how the information about your product will be found in the stores` database when the barcode is scanned.
That depends on how many styles that you have. For example, if you have a birthday card with balloons on the front and another birthday card with a kitten on the front then that means you would need two different barcodes. If you have a birthday card with balloons on the front and the inside says happy birthday and you have another birthday card with balloons on the front and the inside says wishing you a happy birthday then you would need two different barcodes. Remember, the moment you change one thing about your card, it becomes a different product and needs its own barcode number.
You can provide the EPS format of the barcode number to your graphic designer and they can put it directly on the card for you. If you are designing and printing the card yourself, you can put it directly on the card with either the EPS or JPEG format of the barcode graphic. You also have the option of putting a label on the card with your barcode on it.
No, that is not necessary. You will provide that information to the stores where your cards will be sold.