"There is almost no dog." would be different in Korean, but it implies that part of a dog is there, but not all of that one dog. We would say "There are almost no dogs", meaning that there are only whole dogs here, but there aren't many of them. You could also add "There are almost no dogs in sight." using the same structure; you wouldn't use singular unless you are specifically saying "there is one dog" or "there is a dog". Saying "There is no dog." would only make sense in response to someone saying, "Is that a dog?" and it's really a different animal that they mistakenly thought was a dog. I can't think of any other uses of "There is no dog." unless you're waxing philosophical about Schrödinger's Cat thought experiments.