One of the things I love about Rome is the abundance of sidewalk flower stands. Here you can even see one of Rome's classic fontanelle, a constantly-running fountain of cold, fresh spring water that comes from deep underground. These little fountains are stationed throughout the city. To get a drink, you just plug the spigot with your finger, and the water starts to shoot out of a little hole on the top, making an instant drinking fountain. Rome's water is legendary and many say it's the best in all of Europe. So the answer to: "Can I drink the water?" is a definite YES, even though most Romans drink bottled water themselves, probably due to the extremely high calcium content in the tap water. In fact, even with such wonderful water, Italy has the highest per-person bottled water consumption in the entire world, at 183.6 liters per person every year.
Speaking of water, at a restaurant you will be served bottled water, and will have a choice of naturale (still) or frizzante (sparkling). You also may be asked "With gas?" or if you want acqua gassata for sparkling water. I've never seen an Italian ask for tap water, although some of the American students I used to work with would ask me how to say it. In case you're wondering, you can ask for acqua dal rubinetto. I'm not promising you won't get a strange look, but the request will be accommodated.
And of course, then there's the ice/no ice dilemma. In Italy (and most places in Europe I've been to), water and soft drinks are served without ice, but if you ask for con ghiaccio (gee-AH-cho) ice can be added. Drinks are served refrigerated so I suppose they figure ice isn't necessary. As far as free refills and doggie bags go...well, those are still concepts that haven't reached this side of the Atlantic...yet.