I am a singing teacher, and have taught music theory and mechanics of singing, have directed church choir for over 30 years. I have a fairly high level of theory knowledge. I also love to harmonize. Now I am being asked to work with a country band who accompany themselves with ukulele and they want to sing more country harmony. I understand there are 3rds. Perhaps 3rd below, maybe 6th above. And that there maybe 4ths and 5ths related to the chord structure. Is there anything else I would need to know apart from the fact that the harmony voices are dependent on the main voice and are in the key of the song?
Much Country harmony is improvised on the spot, and does usually, as you say, fall into the category of 3rds & 6ths and 4ths and 5ths. I advise the people who are going to be doing the harmonizing to listen to a great deal of recorded country music, especially from the 20s through the 80s, and pay close attention to how the backup vocalists approach their task. Then emulate what they do.
A lot of more recent country music borrows heavily from the rock genre, and so the harmonies don't sound as purely country as one might hope for. Emulating rock-influenced singers is not going to give an audience the idea of what country harmony traditionally sounds like.
One last word of advice. Get your singers to feel free to go variously above AND below the melody line, and not to stay religiously one or the other. This again will give the traditional feel.