The answer to both of your questions is yes.

The Roman schools enforced a very high degree of standardization across the empire into the fifth century. Of course, this was really only enforced on the upper levels of society.

There were also regional dialects, which is partially the origin of the various Romance languages. Spain, North Africa, Gaul, Italy, and the East (which was mainly Greek-speaking) each had their own small variations, variations which grew more pronounced in the lower classes during the later empire.