Can you tell me who (if anyone) is supposed to read the gospel and give a homily at Communion services?
According to Sunday Celebrations in the Absence of a Priest (the most explicit, relevant liturgical document), when a deacon is participating, “it belongs to him to lead the prayers, to proclaim the gospel, to preach the homily, and to give Communion” (Introduction 18).
When a deacon is not participating,
The first two readings are proclaimed by one or two readers. The gospel is proclaimed by the leader. A layperson omits the greeting, “The Lord be with you,” before the gospel.
In the absence of a deacon, the layperson may read the gospel but without greeting the congregation in a clerical manner (“The Lord be with you”). And for the homily there are three options: (a) the lay person may read one prepared by the pastor, (b) the lay person may preach a brief non-homily if he has been given this role by the bishop, or (c) there may be silent reflection. (Introduction 40-41)
Though these are spelled out in significant detail in Sunday Celebrations, they presumably would also apply to weekday Communion services (for which the relevant document is Holy Communion and Worship of the Eucharist Outside Mass).