It's common to split food with people at your table, but usually this applies to antipasti or second courses, rather than splitting a pasta dish. All this said, the world's your oyster when it comes to eating out in Italy, so get what you want and leave the rest.
The waiter gets no extra tip (no tip at all) for bringing you the check. In fact Italian guests don't request the check at the table. They simply get up when they are ready and on the way out pay at the cash register (where the owner usually collect the money) more or less like when you go to the grocery store.
First off, tipping in Italy is neither mandatory nor expected, but if you do decide to do so, the gesture is a very clear indicator that you appreciated the service provided.
In Italy, cash is still preferred in most restaurants, cafes and other establishments, although credit and debit cards are becoming more widely used throughout Italy and are an increasingly convenient way to pay for things. Visa and Mastercard are widely accepted, although American Express is not.