Oooh, finally a question I can actually answer (eh, I'm Roman myself!).

What types of dishes constitute the traditional cuisine of the city of Rome?

Historically speaking, Rome's cuisine has been characterised by its relative "poverty" - i.e. as far as meat is concerned, people used to eat the scraps and residues of animals such as pork, cow or sheep. So you've got delicacies such as the pajata (a veal's intestines), animelle (an animal's belly sweetbread), cow brain, or the infamous quinto quarto - that is, whatever's been left of a cow or a pig after all the more desirable parts had been sold.

Pidgeon, lamb and the like were reserved for important occasions or feast days.

But Roman cuisine also makes use of a limited variety of pastas (alla amatriciana, alla gricia, all'arrabbiata, alla carbonara), soups, and of course the ever-present vegetables - such as the rather notorious carciofi alla giudìa (artichokes fried "in the Jewish manner"); perhaps surprisingly, tomato was not a staple of Roman cuisine and it was used rather sparingly. Things such as the pangiallo were among the few sweets ordinary Romans might have consumed. Local wines from Latium (Frascati, Velletri, Rocca di Papa, etc.) were drank by pretty much everybody regardless of their social standing and boy, didn't people enjoy drinking in the 1800s...

What dishes were consumed by different social classes?

It's not much a question of dishes, but ingredients: the richer you were, the better parts of an animal or the more exotic wines you could afford. Unless we're talking about the feasts of rich people, during which specialties from other parts of Italy were served, your nobleman would've probably eaten the same kinds of pasta a commoner would.

How has Roman cuisine affected that of contemporary Italy?

Not much, actually. Not much. Even its own variety of pizza, or the supplì - which are relatively modern developments - failed to spread in other parts of the country; and the last one's totally unknown outside of Latium.

This said, traditional Roman cuisine hasn't changed since 1870 (but the eating habits of Romans definitely have!).