It is important to note slavery in ancient Rome was not a race based institutution like it eventually became in some European countries and the Americas. Rome acquired slaves from conquering lands, this meant that high amounts of slaves were European, gauls, celts, greeks and slavic peoples. This is addition to northern Africans, Levant and near eastern peoples as well.

Slave's lives varied greatly depending on thier skill and placement. A gallic slave working the silver mines certainly had a harsh and short life and would be more unlikely to sire children one day. However, An educated Greek slave that was a bookkeeper or accountant would have a rather cushy life in comparison as far as slavery goes and would have a much higher chance to be free one day and sire children. Favorite slaves were often freed upon an owners death, and were sometimes given stipends in wills to start a new life as a freed slave.

As far as genetic variances, it would be hard to tell exactly if it was slavery or non slavery intermarriage and interbreeding that would caused ethnicities to intermingle. If a Roman man took a Greek wife, that would give the same similar genetic output as if a Roman man impregnated a Greek slave and created a distant relative. So it gets harder to quantify as opposed to for example the USA or brazil, where we have more identifiable data due to one, the racial aspect of the system, as well as the large geographic distance between Europe and South America. Rome and Greece or Rome and France or Rome and Germany are so close geographically the genetic footprints would be really hard to pin down and would output similar regional data.

However freed slaves absolutley lived lives and had families, jobs and children. They would have been around the Roman culture for so long they would have been able to adapt quickly and integrate rather seamlessly.

Watch Mary Beard's documentaries on ancient Rome, she delves into slavery quite often and has lots of good info on the lives of slaves in Rome.