The Reading of the Apostle (Epistles) is from the lectionary by a lay person. In those churches where there are pews, the congregation sits during the readings of Pauline letters. I want to emphasize that the Apostle is a separate book and is never carried to the altar, and neither is the Old Testament.
Everyone stands when the priest/deacon sings/reads the Gospels. This is in keeping with the Judaic roots of liturgical practice of standing only when the Torah is read.
The Old Testament readings, including and in fact predominantly, the Psalms, is done during the evening services (vespers). The only time when the OT is read (from the beginning until the end) is during the forty days of Great and Holy Lent.
I don't have a Missal handy, so I can't give an exhaustive list, but in the Extraordinary Form of the Roman Rite, there are (I think) 2 or 3 Sundays/Holy Days of Obligation where the "Epistle" was actually a reading from the Old Testament instead of from a NT book. In addition, Ferial Days contain several extra readings from the Prophets in addition to a NT Epistle reading.
I would think (though I'm not sure) that days celebrating Old Testament figures might have "Epistle" readings from the OT as well.