Please excuse my lack of knowledge on this topic. Is non-sexual touch allowed between two people between whom a relationship would be halachically forbidden? I assume the answer is "yes" for, say, a brother and a sister. But does this apply to all cases of forbidden relationships, or just some? (If some, to which doesn't it apply, and why not?) Finally, is the issue of touching a niddah simply not a factor in these cases?
You are mixing up two things that have nothing to do with one another. A man is allowed to touch his close relatives: mother, grandmother, (grand)daughter, and sister (although the last is less recommended, but still allowed). This is always allowed, irrespective of anything else (married or single, nidda or not). (And the same for a woman touching her (grand)son, (grand)father, and brother).
Otherwise a man is not allowed to touch anyone forbidden to him. That means any married woman (aside from his wife) and any unmarried woman who is nidda (and that is presumed about any single woman from age 11 today), since they are forbidden to him. He is also not allowed to touch his wife when she is nidda, since she is currently forbidden to him. (And the same applies to a woman touching a man she is forbidden to, such as any man if she is a nidda and any man but her husband if she is married.)