As it happens, Septimius Severus and Caracalla are some of the very few emperors we have a contemporary painting of.

https://upload.wikimedia.org/wikipedia/commons/5/5f/Severan_dynasty_-_tondo.png

That's Septimius Severus (who hailed from what is now Libya), his wife Julia Domna (from what is now Syria), and their two sons Caracalla (the young boy) and Geta (the body with the scratched-out face...Geta was put under damnatio memoriae and images of him were ordered destroyed).

No one at the time considered Septimius Severus anything other than "Roman." His family included wealthy Roman colonists on his mother's side and native local elites on his father's side. His Syrian-born wife was of an extremely wealthy and powerful priestly lineage of the cult of the sun god El-Gabal (Elagabalus in Latin, later merged with the Roman deity Sol Invictus).

Whether Septimius Severus would be considered "black" today is an impossible question. You can look at that painting and make your own judgement if you want, but the important thing to note is that, while he'd be considered a provincial by the Roman elites, they wouldn't regard him any worse than a provincial from any other part of the Empire, and better than some. Roman provincials had been becoming Emperor since Trajan's day nearly a century prior so even that shift was a done deal.

This answer cannot be answered propperly since the Romans did not have the same concept of race as we do. The criteria that are used in our times to call a person black depend very strongly on colonialist concepts developed in the period from the 16th to 19th century. As a result of this, even if there were dark-skinned Roman emperors we would not be able to tell if they were actually black in a modern contest or, for example, of Egyptian origin with a darker-than-average tint.