It seems like the Normans roamed all around and established kingdoms - sometimes briefly, sometimes permanently - in a variety of culturally, geographically and linguistically distinct areas. In England and Normandy, they established themselves permanently. Around the same time, however, they attacked and conquered Southern Italy. Were these one people? Would they have considered themselves Norman? I know nationalistic identity didn't generally exist in the early middle ages, but what made one "Norman"?
Around the same time, however, they attacked and conquered Southern Italy.
I think you are confused on some of your dates.
The Treaty of Saint-Clair-sur-Epte was signed in 911 and was an agreement between King Rollo (Robert I) and the French King Charles III (Charles the Simple) allowing the Vikings to settle in Normandy.
Now if you are to look at the invasion of Sicily by the Normans, the first contact or Normans on Sicily is nearly 90 years later. Not only that, but the invasion of Sicily wasn't like the invasion of England. The invasion of Sicily seemed to be more or less independent warbands of Normans travelling to Sicily for both trade and financial opportunities (like being a mercenary) and their numbers grew larger as more travelled there over time. It took place over decades, whereas William the Conquerer defeated King Harold in one battle and went on to dominate England within the decade.
It's hard to say whether they were "one people," and I think that is a hotly debated topic among historians. By the year 1000, the Vikings had only been established in Normandy for around 90 years (some had been living there longer, but that is when the treaty was signed so we can use that for a general date) and although there was most likely some interbreeding I would think there were still distinct lineages of "Vikings" and the "Normans" who were natives. It would be interesting if someone with more knowledge on the topic knew if the Vikings would recruit local natives for their expeditions.
One interesting tidbit I found on the wiki page was this
There, Amatus observed that the Norman language and customs welded a disparate group into the semblance of a nation. In 1035, Tancred of Hauteville's three eldest sons (William "Iron Arm", Drogo and Humphrey) arrived in Aversa from Normandy.[16]
It would seem that by 1035 "Old Norman" is around which includes loan words from Old Norse and a similarity to Old French.
So obviously by 1035 there was a level of syncretism between the Norsemen and native Normans.