I will preface that I don't have any sources in front of me at the moment, but I'll go ahead and try to answer this question.

The Normans were Scandinavians who settled in what is now northern France. But, it is important to note that the settlement was not necessarily uniform, and the settlement occurred before the creation of the Kingdom of France. Scandinavians had been raiding and campaigning in Francia for decades, and had besieged Paris twice. The king of West Francia (Charles the Bald?) not only paid off the Scandinavians, but he also granted them the land that would become Normandy. However, we can't make very many assumptions about the early history of Normandy because the original treaty of Saint-Claire-Epte no longer exists, and we don't have many Frankish texts at all from the late 800's and early 900's. There is conflicting information in regards to whether or not Normandy was an independent territory or not, but we don't have much of any evidence to say that Normandy actually owed any allegiance to anyone.

In terms of language, the Normans spoke Old Norse for several generations after the initial phase of settlement, so sometime until about 930 or so (maybe, it's hard to say because of very poor attestations of any language from Normandy at this time period). I want to make it clear that the settlements on the coast of Normandy remained almost entirely Scandinavian for much longer than their fellow settlers farther inland. The Normans began mixing Old Norse with Gallo-Romance at some point between 950 and 1000. However, the Normans were not speaking French. Old French was spoken in the areas surrounding Paris, and not in Normandy. Yes, the two are closely related, but their medieval forms were not mutually intelligible when spoken, and had many, many differences. Norman had a considerable amount of Old Norse vocabulary, completely different phonology (pronunciation), differences in grammar, etc from Old French. Norman was, and still is, not consistent within Normandy, and there are some pretty drastic differences between different varieties of Norman. William the Conqueror did not bring French with him, he likely never spoke or understood French, but he brought the Norman language (or more accurately, his variety of Norman) to England. Norman evolved into Anglo-Norman which was spoken pretty much only in the British Isles).

Anyhow, initially the Normans were just the ruling class of Normandy, but through intermarriage, and spread of their language eventually they grew to make up the majority of the population of Normandy. By the time of William the Conqueror, the Normans occupied most levels of society. In terms of marrying royalty, they were not marrying the French, but they did marry into the English royalty. William had a pretty strong claim to the English royalty, and was actually pretty closely related to the royalty of Scandinavia as well. The story is largely the same for England as well. The Norman Conquest resulted in the Normans becoming the royalty and upper nobility of England, and over time they represented more of the middle and lower nobility, as well as a fair portion of the greater population of England by say 1150 or 1200. However, royalty of other houses replaced Normans in the royal family of England, but the lower, middle, and upper nobility remained almost entirely Norman until about 1450, but by that time Norman identity had pretty much entirely died out in England and everyone was considered English.

Normans were certainly marrying ordinary men and women, but probably not for a few decades after the conquest. What is more evident is the Norman language spreading through all levels of society, we even have letters written by completely ordinary people despite the language not being the native language of most English. Despite the prevalence of Anglo-Norman it wasn't the main language of the court, that would be Latin. Many of documents we have of the language are actually of poetry and medical documents, rather than official court business. However, some very important treaties like the Statutes of Kilkenny or the Treaty of Edinburgh-Northampton were written in Anglo-Norman.

I haven't done all that much research into the Normans of Sicily, but what I can tell you is that it seems like the Normans didn't do too much intermarriage in Sicily, and stayed mostly as royalty. The Norman language did not have nearly as much influence in Sicily as it did in England, and was likely only actively spoken for a few generations before getting conquered by the Holy Roman Empire. The Normans of Sicily went on to marry other noble houses, and had influence on the early Kingdom of Jerusalem. The crusades featured Normans of different factions of houses too, some from Sicily, many of England and Normandy.

Overall, it highly depends on time period and context.