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Changgi
the Wikipedia on General Chinese says that "任" (TCC: gnim~) was pronounced nin in pre-WWII Go'on.
I wouldn't trust their prewar spellings; I've had trouble with them before. My electronic pocket dictionary is quite reliable, and it says that 任 had an original Go'on reading of 'nimu' and an original Kan'on reading of 'jimu'. Their other prewar spellings are generally correct, but for some reason they don't seem to provide the /m/ coda where appropriate. Perhaps because it had been absorbed into /n/ earlier? After all, the mora (拍) 'n' had developed more recently than the others, and words spelt with a final 'n' in prewar orthography were actually originally spelt with a final 'nu'.
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Changgi
Well I haven't read anything related to that... But I just checked Wiktionary. Indeed the Sino-Korean and Sino-Vietnamese readings give an -m ending.
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Changgi
I'm not sure what you meant by "preserving the /p/ coda better" though.
Any historical rule that applied to the /m/ coda applied to the /p/ coda as well, because the latter was considered merely a tonal variant of the former. In this case, Cantonese syllables with labial initials (i.e. b/p/m/f) may not take the /m/ or /p/ codas (e.g. 法 as 'beop' in Sino-Korean, 'biæp' in MC, and 'faat' in Cantonese). Cantonese tends to adopt the /n/ and /t/ codas respectively in their place.
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Changgi
As for help, currently I have the most difficulty in constructing readings where pronunciations don't seem to correspond to each other in Man, Canto and Jp (I use these three as a starting point before referencing the other languages), like 行
The Go'on is gyou and gou, and the Mandarin pronunciations are xing/ and heng/, which correspond to Canto hang4 and haang4 respectively.
I gave "ghiang" and "ghêng" as readings in an earlier TCC version, although I cannot be sure...
They seem to correspond to me...
If a Sino-Japanese syllable (prewar orthography) ends in /u/, it originally ended in /ŋ/ (in most cases). Therefore, 'gyau' ('gyou') and 'gau' ('gou') were originally 'giang' and 'gang' respectively in Kan'on (they are both 'kau' ('kou') in Go'on). For reasons with which I'm yet unfamiliar, the initial /x/ and /h/ consonants in MC became 'k' and 'g' consonants respectively in Sino-Japanese. Thus, they were probably originally something like 'hiang' and 'hang'. Had the prewar spelling been 'geu' instead of 'gyau' for the first Kan'on reading, we could have simply traced it to 'heng', which is practically the same as the MC reading 'hæng' (it really makes me wonder...).
The Cantonese short vowel /a/ seems to be related to the vowel /i/ in Mandarin, probably because Cantonese originally had the short vowel /i/ preceding more than just the /k/ and /ŋ/ codas. Also, the Mandarin vowel 'e' (as in ㄜ not ㄝ) seems to sometimes correspond with the Cantonese 'aa' (e.g. 生). Therefore, my guess is that it was originally 'hing' and 'haang' in Cantonese.
As for Mandarin, 'xing' would have been recently 'hing' (prior to palatalisation), and 'hang' is self-explanatory.
I'm admittedly far from an expert, but these are the patterns I've noticed.