I have read that while the eyes of the deceased were shut before giving the final kiss, the Romans would then open the eyes while on the funeral pyre so that the deceased might look up to the Gods.

A coin was placed in the funerary urn along with the deceased's ashes. I would assume that was derived from the Greeks, as Aulus pointed out after coinage was introduced to Rome from the East. Etruscan and Italic tribes had other ideas about the journey to the afterlife than that a ferryman took you across the Styx and would need payment in coin. The only thing you could compare this later Roman practice with would be the earlier Protovillanovan and Villanovan pozzi found at Rome, where obviously they did not have coins. The explanation is Greek, but it is possible that the practice originated in Rome from some other eastern culture with which they may have had contact.