In "Им хотелось жить по-старому", is старому in masculine or neuter gender? Why?
Presumably neuter. Nounless adjectives used abstractly or referentially take the neuter singular in Russian. (Can't think of the exact term for such usage; what I'm talking about is those cases where Spanish would use lo+masculine, and Latin/Ancient Greek/Church Slavonic would use the neuter plural, if that's any help. E.g. Лучшее — враг хорошего "Better things are enemies of good things.") However, at this point по-старому is, like the previous answerer said, an adverb, and the exact etymology of adverbs is of little relevance. Just how little is demonstrated by the curious fact that the adverb по-русски is technically malformed, being a very old contamination of русски and по-русскому (or the archaic по-русску).