Is the celebrant allowed to read the First (and/or Second) reading, Psalm, and the Alleulia/Acclamation during Mass? Similarly, may the celebrant say the Communion antiphon after the altar has been cleared? Are lay-readers actually a requirement in the Novus Ordo Missae? Similarly, may a priest be the sole distributor of Holy Communion, receiving only the chalice and dismissing E.Ms.H.C.? How much of this is forbidden, set in stone, allowed, or flexible?
All of the things you describe are possible, but not at all to be preferred, with the exception of your suggestion about the Communion antiphon, which should be sung or said at the time of the priest’s communion, not after Communion is over.
59. By tradition, the function of proclaiming the readings is ministerial, not presidential. The readings, therefore, should be proclaimed by a lector, and the Gospel by a deacon or, in his absence, a priest other than the celebrant. If, however, a deacon or another priest is not present, the priest celebrant himself should read the Gospel. Further, if another suitable lector is also not present, then the priest celebrant should also proclaim the other readings.
61 . . . It is preferable that the responsorial Psalm be sung, at least as far as the people’s response is concerned. Hence, the psalmist, or the cantor of the Psalm, sings the verses of the Psalm from the ambo or another suitable place.
While the priest is receiving the Sacrament, the Communion chant is begun. Its purpose is to express the communicants’ union in spirit by means of the unity of their voices, to show joy of heart, and to highlight more clearly the “communitarian” nature of the procession to receive Communion. The singing is continued for as long as the Sacrament is being administered to the faithful.
If no lector is present, the priest himself proclaims all the readings and the Psalm, standing at the ambo.
[100.] So that the fullness of the sign may be made more clearly evident to the faithful in the course of the Eucharistic banquet, lay members of Christ’s faithful, too, are admitted to Communion under both kinds, in the cases set forth in the liturgical books, preceded and continually accompanied by proper catechesis regarding the dogmatic principles on this matter laid down by the Ecumenical Council of Trent.
What you witnessed was not an abuse, insofar as the EMHC is concerned. While the reception of Holy Communion under both kinds is strongly recommended, it is not a requirement. The Communion antiphon, on the other hand, needs to be rechited/chanted as the celebrant communicates.
From the 2002 General Instruction of the Roman Missal (GIRM) approved for the USA, which can be accessed from romanrite.com/girm.html :
“87. In the dioceses of the United States of America there are four options for the Communion chant: (1) the antiphon from the Roman Missal or the Psalm from the Roman Gradual as set to music there or in another musical setting; (2) the seasonal antiphon and Psalm of the Simple Gradual; (3) a song from another collection of psalms and antiphons, approved by the United States Conference of Catholic Bishops or the Diocesan Bishop, including psalms arranged in responsorial or metrical forms; (4) a suitable liturgical song chosen in accordance with no. 86 above. This is sung either by the choir alone or by the choir or cantor with the people.
If there is no singing, however, the Communion antiphon found in the Missal may be recited either by the faithful, or by some of them, or by a lector. Otherwise the priest himself says it after he has received Communion and before he distributes Communion to the faithful.”