Normans?

Based on Lucien Musset - Les Invasions. Le second assaut ...

The creation of Normandy it starts with the agreements between Rollon and Charles the Simple (a diploma from 918 testifies that the king conceded to Rollon pro tutela regni which suggests rather a military investment than a feudal one). The agreements were transformed to vassalage only a generation later, around 940. Though Rollon became part of the Frankish military system, only William "Longsword", his son and successor, made the decissive integration step by giving his Carolingian wife, Liegeard of Vermandois, the territories along the Seine and along the Evreux-Chartres route, thus opening his domain to post-Carolingian influences.
The dukes of Rouen inherited the land of the king and of the Frankish counts, also their ecclesiastical (naming bishops, the custody of the abbeys) and financial rights. The system of rural properties functions in a Carolingian tradition (The author argues that the crushing preponderence of the Frankish forms were justified because they were more advantageous to aristocracy than the Northern customs).

The Scandinavian traits are scarce. Most were eliminated when Normandy took a decissive feudal and Christian orientation. However, there were significant influences in fishing, sailing, shipbuilding, also duke's rights on some categories of shipwrecks. In the rural environment is known only a single property title of Scandinavian influence (the mansloth) and a series of toponyms which illustrate the colonization of the Scandinavians over the Frankish villas. A part of the aristocracy practiced during the 10th century a Northern matrimonial right but limited by the Church. Some of the Duke's laws against criminals bear also a Scandinavian mark (the outlawing - ullac), however even here, the framework is borrowed from the Frankish law. The older Frankish structures (centaines, pagi) revived everywhere (almost everywhere, as unlike in Upper Normandy - closer to the centers of the post-Carolingian authority -, the dukes of Rouen had certain problems in affirming their authority over the Lower Normandy - which escaped also the authority of the last Carolingians).
Unlike in England - where the language is attested by epigraphy in the north-west until the beginning of 12th century - in Normandy there's no evidence that the Scandinavian languages were spoken after 940 (an intresting episode is that around 950, the not-yet-duke Richard I had to travel to Bayeux to learn Danish, unable to do so at Rouen; also nothing points out that it was ever a written language: leaving aside the predilection for exotism, the first Norman works were composed in Latin.), but can be inferred that some groups used it until the 11th century. Especially the fishermen, the sailors, as a some French words have a northern origin: babord, tribord, quille, etrave, havre, marsouin. Other words remained only dialectal like the sea-related "grune" = seabed, the land-related "delle" = piece of land or "londe" = a bunch of trees, also some obsolete judicial terms: "gaives" - goods with no master. Otherwise, the vocabulary was little altered.
In art, their influence in Normandy was insignificant (unlike in England). The artistic revival from 11th century is either continuing a Carolingian tradition or presents spontaneous developments (some similarities between Normandy and Scandinavia are due to a reverse influence, the Romanesque art of Normandy influenced the Christian art from the north, especially in Norway and western Denmark).

As for the ethnicity of the colonists, according to their onomastics, it seems the majority was Danish and a large part of them inhabited for a while the north-east of England from where they brought new/altered anthroponyms, but also agricultural terms, especially betwen Bayeux and Orne, and less in the north of Cotentin. Other colonists, after inhabitting a while in Ireland and Scotaland, came in Normandy wearing names like Murdac or Donecan. These colonists have a Norwegian origin and estabilished mostly in Cotentin. The women were in a severe minority (the analysis set having 3 such names comparing to around 80 male names), so the newcomers must'd married local women since their first generations.
The duke's family seems Norwegian, though sources credit Rollon both as Danish and as Norwegian (but the sources of that time hardly made the difference). Anyway, it was not unusual for a Norwegian chief to rule over a Danish army (the Danish city of York had its Norwegian kings). An interesting aspect that also supports the Norwegian origin of Rollon is a certain duality (Norse/Frankish) of their names encountered during the 10th century: Rollon was Hrolfr and Robert, his daughter was Gerloc/Geirlaug and Adelis, the wife of Richard I was Gunnor/Gunvor and Albereda. 
As for the geographical distribution, two regions knew larger colonizations (and consequently influences): the north of Contentin and the western Pays de Caux. Bayeux, Roumois between Seine and Risle, Avranchin, Bocage Virois were less affected and the fields of Eure and the lands of Bray were almost untouched. Auge and the valley of Seine were scarcely colonized (a note though, not all the settlements marked by Scandinavian toponymy were colonized in that age. In some cases there are Scandinavian lords whose names were integrated in the place's name: Mondeville = Amundi villa, Roumare = Rolmara. In other cases there are dialectal borrowings which generated toponyms years after the periods of colonization like the aforementioned "londe").

Two social classes was altered: the aristocracy and the sea-shore populations. Nowhere the Scandinavians behave like a closed group. From the beginning of 11th century the aristocracy was extremely composite: besides Danes and Norwegians included Franks, Bretons and even Germans. The indigenous elements altered it greatly thus no significant differences were between the conquerors and the conquered at the beginning of 11th century.
Several other Viking bands came later to Normandy (the so-called second wave). The dukes of Normandy used them for their own interest: against Louis IV in 945 or against Eudes of Chartres in 1013 (an action already mentioned in this thread). However, the Vikings unable to integrate in the new structures were exiled. These ones, still bearers of the Viking culture, had a considerable role wherever they fought, in Spain or Italy. They are among the founders of Norman Italy.
Normandy also had a direct gain from some Viking loots which were directed to Rouen. However, the dukes of Normandy didn't want the Danes to have full control on England and they supported the anti-Danish movement while Ethelred II and a handful of followers were exiled at Rouen. It seems that this is the true origin of the action taken in 1066 which were not especially against the English, but against the Anglo-Danish aristocracy represented by Harold.

One last point is their double-faced identity. Snorri Sturlusson, writing in the 13th century, calls the old dukes of Rouen "Rudhu jarlar" (the jarls of Rouen) while Richer, hating Normans, calls Richard I "comes piratarum" (so like I thought, it was an insult. However I think it's related to Richard I's ancestry and the fact that his "cousins" were still haunting the seas).
But from the "official" point of view they were French/Frankish: the diplomas call them comites Rothomagenses (a title for the counts of Rouen, according to the Carolingian tradition) or marchiones (as marquises, their duty was to protect Seine from pirates - i.e. Vikings!).
As you can see the title listed here are almost perfect opposites. Though Musset doesn't suggests it, it seems to me that their Northern titles are relevant only as mythical reflections, it's about one late Northern account (which barely translates their Frankish title by jarl while we know how were they integrated in the post-Carolingian system) and one venomous remark reflecting a personal aversion.