The article can be ommited because you're talking about sth general (not a specific kind of credit card) and it's implied that it is supposed to be only one.

You could say Πληρώνω με την πιστωτική κάρτα, without ommiting the την, but that would make it just a bit more specific, wouldn't it? I hope I helped. ^.^

Yes, in the sense of "next to" -- "I'm waiting for you by the post office", for example, when you are standing next to it.

But you're not paying next to the credit card, you're paying by means of the credit card, with the aid of a credit card.