Are the Game of Thrones type societies that are usually associated with medieval England accurate portrayals of what their class system was really like?

No.

The problem with using GoT as an example is that it borrows from across several hundred years of aristocratic life to create an image which reflects none of them accurately (it's like a house of mirrors). So these aristocrats hold land like they were unimportant tenants of the eleventh-century, they act like lords from the eleventh- through thirteenth-centuries, but adopt a titular and hierarchical structure similar but not accurately reflecting the fourteenth- and fifteenth-centuries.

This essentially makes it impossible to answer easily as I would have to disassemble the features of lordship and society in GoT and then identify which period they come from. Something no sane person not writing a Ph.D. on ASOIAF would ever want to do.

Were there a variety of noble houses that existed underneath a unified king?

Well, the last bit doesn't really make any sense. Are you asking if there was such a thing as a kingdom of England?

It might help if you step away from the term 'house' and instead think 'family'. Our word 'family' comes from the Latin familia (household), so it's not a major jump:

Were there a variety of noble families in medieval England?

Yes, of course. These families might attach themselves to a name but the name was not usually a 'last-name' it was a 'place' (unlike GoT, the Starks would actually be the Winterfells - or Winterfell would be called Stark). These were things like Geoffroi d'Anjou (Geoffrey of Anjou). It was a development of the twelfth-century that really began attaching importance to these descriptors (often combined with proto-heraldic insignia, Geoffroi's lion was to become a lasting symbol of the English crown). This is the reason there are an abundance of Norman placenames among the aristocracy of England (great and small), take for example, the Nevilles. By the later Middle Ages the major families were still taking their names from locations, if only to differentiate themselves. The Beaufort family are a good example.

What initially differentiated a noble from a peasant or commoner?

One was free, the other was not. One was a warrior the other a laborer.

how did this class system arise?

See my response here, this is really a bit too big to set out right now. Class is a rather ill-fitting term beyond the distinction between free and unfree in the period up to 1200. After that point class begins to solidify among the aristocracy into more identifiable ranks.

How much would the nobility of the far north know of the nobility of the far south?

England really isn't that big. It probably wouldn't take little more than a week to travel across England even with an army in tow. If an aristocrat was a sheriff or bailiff for the Crown then he would have had to report to the Exchequer twice yearly (Easter and Michaelmas). They might serve with the king on campaign, or, as I have said, visited their lands across the country or the king's court in London. Kings often held colloquia (conferences) with their magnates as these were an essential feature of medieval lordship and governance, and of course in England these could be found at Parliament. There were lords who wanted to keep themselves to themselves (or understandably didn't want the added hassle of acting as a sheriff or bailiff), but these were somewhat exceptional.

Would a Northumbrian baron know what a Marcher lord had for breakfast? No, but then why would he care? He might well know that the Marcher had killed another Marcher or that he was suing the Earl of Chester in the king's court, but these are important issues.