Yes they have at least starting from the late republican age, there are variations for men and women, from the republic to the later empire and other really interesting but really complicated situations, just remember that every respectable house had a small altar with clay or wooden figurines of the owners ancestors who were supposed to give protection and was given daily offerings. This to draw a better picture of how much family was important among them, even slaves who had only a single name when freed usually got the other two names from their former owner.

Every ancient roman citizen born or made free from slavery had at least three names, "praenomen" which stands for our first name and was given at birth by parents, a "nomen" which roughly stands for surname or family name and was used to identify the descent from a noble "gens" but was also used by plebs and came from geographical origin, ancient Rome founding tribes, etc. And finally a "cognomen" based on a nickname that was given for whatever reason from joke to celebration, famous historical figures are known for this, Caligula came from caliga, leathers sandals used by military personnel who gave the young aristocrat growing among the legions this nickname literally meaning small sandals. They were also given for celebrating an achievement like Scipio called Africanus after defeating Chartago, also the cognomen was sometimes used to identify different branches of the same family so his descent brought the nickname proudly for generations. As I said before it's a fascinating and extremely complex matter of study with plenty of documentation given by epigraphy in catacombs and other burial sites, the study of roman names and their origin was and still is a lifetime mission for respected historians and that's why I answer your second question with a humble I don't know, I am just an amateur but I hope you'll find some useful information on common names at this link and surnames here.