This answer is probably more appropriate for Clans from the late 16th Century to the 19th Century but I'll address how they changed from those in the 14th Century and earlier.

The Clans in the Scottish Highlands were organised in a semi-feudal system with the Chief being at the top of the hierarchy. The Chief was a feudal landowner and most Clans originate from the granting of land under Scottish Kings such as David the 1st and Malcolm Canmore. The men that received this land then passed it down to their sons. This is where you often get clan names. Mac meaning "Son of". So the Chiefs owned the land and in earlier years would often live on that land close to their tenants. They would rent out the land to their tenants who would pay their rent through the goods and services they provided.

By the late 15th and 16th Century a new class was emerging in the Clan system. The Tacksman. The Tacksman became the managerial class of the clan system. They get their name from the land that they rented. As Chiefs began to live further and further from the rest of the Clan they would rent out the land in Tacks - Tacks were an unmeasured piece of land. The Tacksman were the class of men that rented this land. It was their responsibility to manage the land, generate produce from it, taxes and to provide men for war. The Tacksman would sublet his land out to the other tenats and would be responsible for collecting taxes. The Tacksman were often related to the Chief being either younger sons, younger brothers, cousins or other distant relations. But some were also Chiefs in their own right, renting land off of bigger clans. So The Tacksman could claim to be nobles. The quality of Tacksman varied greatly some helped to work the land while others sat back and enjoyed a more comfortable life. The Tacksman tended to live in the Townships with their tenants.

The last Class were the subtenants the people that worked the land and created the produce for the clan. These people rented their houses from the Tacksman for medium terms ranging ranging from about 10 to 20 years on the most part. How much rent you paid determined how much land you got to grow produce on. Most tenants by the 16th and 17th century were living in Highland Townships or "Baile" in Gaelic these were small collections of houses surrounded by land that would be used for growing oats and barely and sometimes rye.

To understand the Clan system properly you have to understand the relationship they had with Cattle especially in relation to your question about inbreeding. The Highlands are a much hardier landscape than the rest of Britain and that made the growing of produce much more difficult. Cattle became the main industry in the Highlands. In Summer the cattle would be taken to the "Sheilings" - the small structures located high in the hills beside where the cattle would graze in the summer - and in winter they would live in the houses in the townships. about half of the township would go up to the Sheilings with the cattle and this is were much of the young people would court each other as people from different townships would sometimes share the Sheilings for their cattle.

The size of the townships was also important. Townships varied in size but some could have populations of up to 300-400 people. So the likelihood of inbreeding was low. Also marriage within families wasn't uncommon - not sibiling to sibling but among cousins, aunties and uncles - There was most likely enough genetic diversity to stop major genetic diseases.

While the Highland township do have some similarities to other feudal villages there are also many differences. The Tenants for one are not serfs. They are not tied to the land like a peasant in Medieval Scotland or England would be but are free - but landless. They are of course restricted by poverty and distance so most tended to live in the same location for generations. While they couldn't just go into the woods and build a house they could have a small choice in where to live - This isn't to be romanticised, this freedom in reality was non existent for most people due to the poverty and hardship they lived in, moving away out of self choice for most people would never be a reality. Like other feudal societies the townships would keep animals. These would be Black Highland Cattle, Chickens and an extinct breed of sheep. The land for growing produce was on the whole divided in two with "Infields" and "Outfields". The Infields were the most arable land and most of the oats and barley consumed by the township would be grown here. The Outfields were used mainly to grow oats and barley for the cattle, or surplus for market. Although very often the outfields would have to be used for Human consumption given the frequent famines at the time. These Infield and Outfields were further divided by the people that rented them. The system was known as the Runrig system and Tenants would be allocated rigs of land - some Infield, some Outfield - depending on the rent they were paying. Some tenants would have no rigs due to the low rent they were paying and had to make up their rent by other means, mainly looking after cattle, collecting resources and providing services such as metal work, wood work, weaving and hunting.

When It comes to hunting I'll use Badenoch as an example as it is the area of the Highlands I am most familiar with. In the Townships in Badenoch hunting was mainly used for trade. Most of the animals that were caught would be sold off or used to pay rent. In Badenoch in the 16th - 19th Century they hunted ducks, geese, grouse, hare, fish (but not Salmon as it belonged to the King) wild boar and other small game. Deer and Salmon belonged to the King so hunting it was classed as poaching - although it was rarely inforced. Other animals often associated with the Highlands such as rabbit and pheasants hadn't been introduced to the area yet and wouldn't come north until the 1860s.

So In Summary by the end of the end of the 17th Century the Highland Clan system had established a 3 Class (4 if you Include the King) system. With the Chief who owned the land on the top, who rented his land out in Tacks to Tacksman, who then sublet that land out to the ordinary tenants. Their main produce was Cattle, Oats and Barley. And while most of the clan claimed at least some noble blood, we can only say with a degree of certainty that the Chief and the Tacksman would have any. Inbreeding wasn't a problem on the most part due to frequent interaction with other townships and the size of the townships.

While as I have noted this is more appropriate for a later period. The main differences I have come across in my research have been: The emergance of the Tacksman, The increasing importance of cattle in the 15th and 16th century and the tendency for Chiefs to move away from the Clan and into castles.

If you'd like more Information just ask.

Sources:

James Hunter: Set Adrift upon the World

I.F. Grant: Highland Folks Ways

James Hunter: Making of the Crofting Communities

EDIT: Expansion of the growing of Crops

Follow-up question: how did clan tartans arise? I was at a tour in Edinburgh yesterday where the guide said tartans are actually relatively new traditions (19th century) developed to impress the English with a false level of culture and tradition. Is that true?

Lets extend your question out to the 16th century (other answers cover the 18th and 19th).

Taking your questions:
Descent from a common ancestor/founder:
Every Clan had their origin myths with varying degrees of historical accuracy. Clann Domhnaill for example (Clan Donald) claimed descent from Aonghus Mór mac Domhnaill (the great grandson of Somerled - "Ri Innse Gall" or King of the Hebrides" - i.e. Aonghus mac Domhnaill mac Raghnaill mac Somairle mac Gilla Brigte) - other branches of Clan Donald claim a different descent from King Duncan I - and the stories would be reinforced by the clan bards retelling the stories of their famous ancestors (and Somerled certainly counted there).

The main surving source for the origins of the clans is a manuscript (known as MS 1467) and edited (poorly) in Collectanea de Rebus Albanicis and (somewhat better) in Celtic Scotland. The origins (pedigrees really) are really about ensuring the appropriate length and quality was on display so that appropriate respect would be given (see Sellar's account of the the feast at Aros on Mull and the effect failing to properly provide the respect for pedigree could do). The TLDR of which is that John MacDonald took the opportunity to seat the MacDonald chiefs in attendance for dinner. After seating a number of key chiefs and bards in their proper order, he sat down saying he was the best of the rest and they could fend for themselves. Those not seated stormed out swearing revenge at the disgrace).

Additionally, from the fifteenth century on there is a change in direction for many clans who switch their pedigrees from a Celtic (or Britonic) source to a Norman ancestor instead. The Campbells (among many many changes) changed from being descended from Mervie son of King Arthur (or sometimes Mervie is Merlin) to that of Beauchamp in France (via the manipulation of cam beul to De Campo Bello).

The David Sellar article is very instructive in the review of the tall tales!

Marriage:
The clan can be seen to cover the very locale specific (MacDonald of Glencoe) out to the large confederation (e.g. Clan Donald and Clan Chattan). "Dynastic" marriage could have specific benefits whether in land or in alliances. There is the example of Colin MacKenzie of Kintail's marriage to the daughter of Grant of Freuchie which, following the marriages of his sisters, resulted in him now being brother in law to six other clan chiefs. Three of his daughters married other chiefs. The lower in the strata you go there would likely be a similar effect with looking to marry "over in the next glen", but as marriage records are non-existant/sparse until much later this would be hard to document for the lower levels of society (a bit like proving that there was no intermarriage in a small village in the middle ages).

Fealty etc:
In many ways the middle ages in the highlands sees a mixture of feudal society and Celtic society. One aspect of Celtic society (and indeed law) that persisted was that the Chief had a responsibility to his people. If he failed in that responsibility he could (and would) be replaced. Correspondingly his people were to owe him absolute allegiance (which depending on the period would exclude the King). and we can see this in surviving indentures and bonds of man rent (see sources). Chiefs held absolute authority over their lands and held the right to manage law and order within them (whether through Baron courts) or in more ad hoc fashion. Indeed the Lord of the Isles held a periodic "clan council" where fourteen representatives of the clan (septs etc) met with the Lord of the Isles at Finlaggen on Islay to review and decide issues of importance for the clan as a whole.

Surnames:
No, this doesn't really come in until the 18th century (despite what Dr Johnson made upwrote. In Celtic society the patronymic is what you are known by (whether Chief, tacksman or clansman (at least locally). If we take Clan Cameron as an example. The first mention of it is in the confirmation of a 1472 charter Register of the Great Seal of Scotland October 24 1495) with the Chief listed as Alano Donaldi Duff capitaneo de Clan Camroun (i.e. Alan MacDonald Dubh- or Black Haired Alan Son of Donald). Other charters and land grants which are local in nature show patronymics while those drawn up outside the area will show a different name (there is a classic example whose source escapes me where the same individual is referred by different names in charters depending on where they were drawn up - in this case locally in Argyll he has a patronymic but when the same charter is confirmed he is listed as a Campbell.

Sources not directly linked:
R.W. Monroe - "The Clan System - Fact or Fiction" in The Middle Ages in the Highlands
David Sellar - "Highland Family Origins - Pedigree Making and Pedigree Faking" in The Middle Ages in the Highlands