Is it rude to ask for parmesan cheese on seafood pasta?

Yes. It is considered a culinary sin to combine cheese with fish or seafood dishes. Do not ask for it.

Do I have to pay for water?

Yes, you will be charged for bottled water (naturale or frizzante). If you want tap water, you must specifically ask for 'acqua del rubinetto,' though many restaurants may refuse or steer you toward bottled.

Dining Like a Local: Italian Table Etiquette

Italian dining is about the pace and the quality of ingredients, not complex rules. Keep your hands visible on the table, avoid ordering cappuccino after 11:00 AM, and wait for the host to say 'buon appetito' before you pick up your fork.

Respect the sequence. Understand the order: Antipasto (appetizer), Primo (pasta or risotto), Secondo (meat or fish), Contorno (vegetable side), and Dolce (dessert). Don't ask for your salad to be served with your pasta; it usually arrives after the main course to cleanse the palate.
Handle the bread. Use bread to 'fare la scarpetta' (soak up extra sauce on your plate) only if you are in a casual setting, and do it by holding a small piece of bread with your fork, not your fingers. Never put butter on your bread; it is eaten plain or with olive oil.
Order drinks correctly. Coffee with milk (cappuccino, caffè latte) is strictly for breakfast. If you order a cappuccino after a meal, the waiter will know you are a tourist. Stick to espresso (caffè) or digestivo (limoncello, amaro) after dinner.
Master the check. In Italy, the check (il conto) will never be brought to you automatically. You must catch the waiter's eye and ask for it by saying, 'Il conto, per favore.' Tipping is not mandatory; leave 1-2 Euros for a coffee or round up the bill for a meal.

Is it rude to ask for parmesan cheese on seafood pasta?: Yes. It is considered a culinary sin to combine cheese with fish or seafood dishes. Do not ask for it.
Do I have to pay for water?: Yes, you will be charged for bottled water (naturale or frizzante). If you want tap water, you must specifically ask for 'acqua del rubinetto,' though many restaurants may refuse or steer you toward bottled.