In Anglo-Saxon England, the "earldorman" was a local who administered a shire or group of shires for an absent Scandinavian Jarl/Earl; in exchange for leading any forces levied from the area and holding court with the bishop, the earldorman was given the "third penny" (one-third of any fines paid in the shire) and held an estate. While the position wasn't inherited like a title today, it did tend to run in the family - and often these families were related to/descended from the pre-Scandinavian rulers of the area. The ealdormen became "earls" shortly before the conquest, and following it they were known as either earls or counts to the Normans. As the administrative duties passed to men of a lower rank - viscounts - earldoms were left as hereditary aristocratic titles, the highest in the land until 1337, when the position of Duke of Cornwall was created. (The first English marquessate was also created in the same century, and ranked higher than an earl but not as high as a duke.)

Being an earl did not grant any specific benefits during the period of time you're asking about, and countesses didn't play any specific role separate from other aristocratic women. By this time, earls were even being created without any relation to actual counties or areas - that's why you end up with titles like "Earl Ferrers" and "Earl Spencer" among all of the earls of [placename]. It was simply a title handed down from aristocratic fathers, or given out by the king to viscounts who were felt to deserve a promotion for some reason. So your question is not quite answerable!