Did Ancient Rome have any restaurants? Places where people would sit and order food and socialize? If not, then when did restaurants become the norm?

It did! These were called thermopolia, and they were ancient Roman businesses that served the public both hot and cold food, usually over counters that look strikingly similar to the modern day deli. Examples of these early restaurants can be found all over Italy, with an excellent example of a preserved site found in Pompeii's Thermopolium of Aselina. Note the deep vats in the counter.

While it's not unreasonable to think that these early restaurants would mostly cater to the upper classes, this actually wasn't the case. Thermopolia were most useful to the relatively poor, who couldn't afford homes with kitchens, making a place to meet and enjoy drink and hot food an important aspect of the lower-class Roman social life. This also gave them something of a dubious reputation in high society. (By Jupiter, imaging sharing a meal with a Greek. Or an actor.) But the site of Pompeii alone contains 150~ known thermopolia, so they were definitely widely patronized.

Archaeologists disagree about some of the other features of these establishments. Rooms above some of these cook-shops could've served as something akin to an inn, or they might've been chambers used as a brothel of sorts.

I would love to see just how far one could trace these establishments back in history. The existence of so many around the turn of the millennium makes it hard to believe they were a relatively new development.