Hi, "I have got a mother." You need the indefinite article. Similarly, "I have got a big family: mother, father, ... and me." Of course, if they were the subject of the sentence, it would be (for example), "Mother, father, ... and I live in Moscow."

"I have got a mother." There are many mothers, and your mother is one of many: a mother. Doug has answered your second question well. :)