Does anyone know where I can find the canons on how the Orthodox Church "penances" an Orthodox Christian who has children outside of marriage, lives with a man (or woman) for awhile but then gives up that relationship, or can give me a link to the pertinent info?
Canons (especially in the plural) do not discuss this specific issue. However, you should look up Nyssa 4 and Basil 22, 25, 26, 38. Those would be the most relevant ones. Basically, they say that anyone who has sex outside of marriage should have three years of penance, but both Basil and especially Gregory of Nyssa emphasize that the confessor must adjust the sentence depending on the person's contrition. So, ultimately, it is all up to the confessing priest. A lay person cannot give themselves a penance.