Is it normal for my baby to have swollen genitals?

Yes. Newborns often have swollen genitals from mom's hormones and sometimes related to the trauma of coming through the birth canal. Older babies should not have swollen genitals. Suddenly swollen genitals in combination with fussiness or definite pain should be considered a medical emergency.
Usually not. Unless there is a known problem, infection or recent circumcision then this is not normal and should be seen by a doctor asap. A boy or a girl with swollen genitals is not normal.