Sorry, I can't get anything other than what you already have. The meanings are slightly different
御謌所 / 寄人 The first three are an official title. Something like "Official Court Poet". The second part is just saying "person". So the two together are "Member of the Official Court Poets"
須川信行 Nobuyuki Sugawa 1839-1917 (physician who turned full-time poet and, as above, one of the "official court poets".
翁懐紙 Writings/notes of an old man. (Self-deprecating, I think. Like "the scraps of writing of an old man"
月前碁 poem title: go in the moonlight ? ← Yes, you have it right.
interior:
癸亥ハ月上浣 Japan uses the Chinese zodiac cycle as well. I Japanese we'd say "Mizunoto August. 1st third of the month." Could be 1863. Could also be 1803 or 1923 or 1983. The question would be, when did Nobuyuki attain the title (or the sufficient skills to be recognized as) Official Court Poet. 1863 might be too early, as he was only 24 then.
襄陵簽 I'm not sure about this. Jōryōsen is the probable pronunciation, but not sure of the meaning. A name, i think, and possibly the name of the person who presented the box. Its not a real name, but a made-up "pen name".
seal = the seal reads 山襄之印 (seal of Sanjō). This is related to the above. It would be the personal seal of the person who is using the name Jōryōsen.
御歌所信行 Nobuyuki, of the Official Court Poets.