As already mentioned, the smallest possible sum is 27, resulting from the numbers 1, 2, 4, 5, 7, 8 arranged like this:
Optimality is a consequence of the observation that no three sides of the die may show three consecutive numbers (e.g. 1, 2, 3 or 3, 4, 5 are not possible).
To see this, suppose that $n$, $n+1$ and $n+2$ were all on different sides of Enrico's die. Then because of rule #3, $n$ and $n+1$ must be located on opposite sides, and so must $n+1$ and $n+2$: a contradiction.
The fact that no three sides may show three consecutive numbers implies that there must be at least two gaps in the number sequence between three pairs of consecutive numbers, and we see that the sum of the sequence is minimized for gaps at 3 and 6.