I know limbs have their compositional rules. I'm working on a self-portrait where I'm contemplatively looking left. To emphasize the effect, I want to crop my right ear off (hey, no Van Gough jokes!). Is cropping out someone's ear a compositional no-no?
It depends on the overall image. If you have a single, tightly-framed face, I don't think there's any problem. If it's a wider image with a single subject at the edge and something balancing the image on the other side (even if just white space), cutting the ear might be a bit strange, but that strangeness could be part of what the composition work. That sounds like it's probably your case.
On the other hand, if it's a posed group portrait with multiple subjects, I'd avoid cutting off the people at the edges. That mostly looks like you just didn't have enough space for the shot, or you're cutting out something to disguise a mistake. This is especially true if it's just one poor person who gets the knife, but the same would apply if you do it to a person at each edge. But again, it really depends on what's going on — if it's a dynamic scene where someone looks like they naturally moved there as part of some ongoing action, I think we'd be back to "no problem".
Yes and no. All rules photography are there to be broken. They're just guidelines. If it looks like it works and is conveying the mood that you are trying to achieve, then go for it
Again, depending on your specific image, at a guess I'd say cropping out an ear will ad tension, uncomfort, etc.