1. At race tracks, the payout for a winning horse is posted on a board. That is what is meant by "numbers." It is the payout if that horse wins. I do not know anything about Italian race tracks, but in America the numbers are either written as something like 6/1 (meaning you get $6 for every $1 you bet) or something like $14.20 (which means you get $14.20 for every $2 (why $2, I don't know...) you bet).
To me, 18.50 *looks like* the profit paid out for every 10 lire a person bets. So if you bet 100 lire on that horse and it wins, you get a profit of 18.50 * (100/10) = 185 lire. Again, I am not sure how Italian race tracks work, but it is clear that 18.50 is the payout or the odds, and "the numbers" (in "posted the numbers") refer to all the payouts/odds for all the horses competing.
2. A ten-lira bet means simply that. A bet of 10 Italian lire.