So, a few things to unpack here. First, there were no dukes in England until the 14th century, and no counts at all, though earls corresponded in status to continental counts. Second, there were only a handful of counts earls at any given time in the 11th century: never more than ten, and usually less. They were major landowners, endowed with their titles directly by William the Conqueror as a reward for service and support, especially during the Conquest. Third, earls were less politically powerful than on the continent, as English holdings were, by design, broken up geographically. You might be the earl of Cornwall, but hold land all over the kingdom and in Normandy itself.

I'm grossly oversimplifying, but the social structure of Norman England basically went king -> tenant in chief -> sub tenant -> peasant. Theoretically, a tenant in chief was someone who held land directly from the king and had no intermediary liege, while a sub tenant held land under another noble. Not all tenants in chief were titled earls; many were simply large landowners, and are often called barons. Sub tenants could vary wildly in how much land they possessed, ranging from substantial landowners to relatively poor knights who owned a manor or two.

So with that said, your relationship to your property would depend greatly on how much you owned. If you only owned a few parcels, you probably lived on one of them and interacted regularly with the residents. You likely didn't manage the property directly, but worked through a bailiff, basically a peasant-manager. If you possessed holdings scattered throughout England and Normandy, it was less likely that you interacted regularly and in person with all of them, though you would have traveled between your lands periodically. You probably worked through intermediaries much of the time.

Likewise, your daily routine would depend greatly on your social status. A lower-ranking aristocrat, such as a knight or petty lord, had more obligations. He had duties on his land, including hearing disputes from his subjects, functioning as a petty judge. He would spend a portion of the year doing castle-guard, that is garrisoning one of his lord's castles. In event of war, he would bring his men, which might be only a squire or two if he was poor, to fight with his liege lord. He would also spend some time living with his lord, especially when he desired his counsel. A great lord's duties would be similar, but at a higher level: attending to the king personally, leading his household to war, and settling disputes among his sub-infeudated lords.

But life was far from a series of onerous obligations. The aristocracy, great and small, shared a lifestyle and hobbies. Hunting was probably chief among these. This was typically not a single guy stalking through the forest with a bow and arrow, but a rather grand affair, with friends, relatives, and retainers riding on horseback, servants afoot, and a great many hounds ranging about in search of prey. Tournaments were in vogue, moreso in the 12th century than the 11th. This was not only an outlet for displaying martial prowess, but a good opportunity to practice for war (tournaments in the 11th-13th centuries were hyper-violent mock battles between teams of mounted knights) and, if you were skilled or lucky, to make some money by ransoming defeated knights and their possessions. Feasting was another diversion, both hosting and attending the feasts of greater men. The aristocratic class largely became literate in the 12th century, and afterward it was trendy to write poetry and songs.

Edited because I'm dumb and wrote count instead of earl, invalidating the point I had just made.