There are no stupid questions.
I don't think the cough syrup would help one fall asleep, at least at the very small doses that I take.
A single dose of melatonin probably would not either. Melatonin apparently is helpful in resetting our circadian "clock" in the case of jet lag.
It worked for me last Spring when I travelled to Greece, Turkey and Israel, especially on the return to the US from Israel.
I continued taking the DM syrup through that time and did not notice any effect of combining DM and melatonin.

Robert