As for face-claims, perhaps the God/Goddess has one preferred image that they use pretty much all the time, and just alters it in order to do some frick-frack with mortals. Or non-mortals, depending on which myth you're looking at.

Either way, I'm sure they settled on a final image - for example, Hermes is nearly always portrayed as a young man; Hephaestus is always portrayed as a disfigured dude, and Zeus always has his rockin' beard.
 
So, I like the idea of having F/C for their everyday image - at the very least it gives us an idea of what they may look like.
Obviously, if people share a Godly parent, you'll have to debate over it, but I highly doubt it'll cause many issues.