We have no tradition or information concerning whether Mary was baptized or received any of the other Christian Sacraments. She was raised and married according to the Jewish religion, and she received the standard Jewish rite of purification for women who have given birth, using a mikvah (bath) like those that exist to this day to one side of where the Temple once was. Beyond this, there is no record.
Because she was sinless, she did not need Christian Baptism. Jesus did not need Baptism, either, but he did accept John’s baptism of repentance on behalf of those who would believe in him. (According to patristic tradition, this act spiritually sanctified the water that is used in Christian Baptism, so that it would be efficacious for regeneration.) It is possible that Mary, too, received a baptism of repentance (not the Sacrament) in the spirit of her Son. But again, there is no record that she accompanied Jesus to the southern area of Judea where John the Baptist was preaching.
The idea of Mary receiving Communion is rare among either Catholics or Orthodox. I suppose it is possible, but you outline a theological objection; she already had the fullness of divine grace in view of her divine motherhood, and no Christian Sacrament, not even the Eucharist, could make her more holy. Personally, I am inclined to think that she did not receive Communion but continued to follow the Jewish religion, at least until the day when the Holy Spirit descended upon the group of believers in the Upper Room. Scripture states that she in the care of the Apostle John and was present at the meetings that were held there after Jesus’ resurrection.