(1) Was there a special rationale for assigning people to one over the other?
This “fork” of the family would have been inevitable because of population growth inside the imperial family. They must have wanted to get this done gently. The solution was to give a new family name and to designate as the local official.
Prior to this era, people already suffer from high tax caused by growing number of government officers especially after “大宝律令”. It is said that many people left their designated fields due to heavy tax (“浮浪人”). To make matters worse most officers aggressively committed corruption (as can be seen from installing “勘解由使”).
As the result the government revenue faced big risk. Lack of financial resources and the “position” forced the government to decide ”fork”.
(3) Did they provide/require special services?
They were assigned as 国司(Kokushi). Many such local officers are needed to keep heavily centralized government.
Interestingly heavily centralized government got stuck due to reasons described above. The power of government had to be decentralized. As the result Kokushi residing local and their offsprings were gradully taking over the power. One shocking case was 平将門の乱(Conflict by Taira no Masakado). As seen in the 平(Taira) name, 平将門(Taira no Masakado) was a child of a Kokushi who was given that name during the imperial “fork” event.
Many local powers rise and fall and Japan stepped into the era of Arms and Samurai.
(2) Were clans supposed to be the overriding social identity for their members?
(4) For example, did Saga and Seiwa Genji both identify as Minamoto and were therefore more likely to cooperate with each other than with other clans?
I think some kind of “social identity” certainly existed but it was not always important.
First of all, 源(Minamoto) or 平(Taira) belong to 朝臣(Ason) which is the second highest rank of かばね(Kabane). Minamoto and Taira are supposed to be family names, while Ason are kind of social rank of family names. Such association of family names and ranks seems started since ancient era.
Therefore having those family names definitely showed the social rank in this era.
However, descending generations, most of family names of nobles and samurais shrunk to just handful of kinds. We often refer to “源平藤橘”, which represent first 1 character of major family names at that era, 源, 平, 藤原, and 橘.
It’s better to take 源平合戦(Genpei War) to discuss this topic. We explain this as the war between 源氏 and 平家, NOT between 源氏 and 平氏.
In this context, 氏 sounds just like “name” and it doesn’t necessarily represent kinship. Whereas 家 means like “house”, “family”, “kinship”.
源氏(Gen-ji) represented by 源頼朝(Minamoto no Yoritomo) rebelled against 平家(Hei-ke) represented by 平清盛. 平家(Hei-ke) was the establishment at the time and it consisted of kinship (it happens all the time). In the mean time many 平氏(Hei-shi) joined to the resistance group which is regarded as 源氏(Gen-ji). That means many people who have Taira name wern't side with the establishment Taira group. That’s why we use the term 平家(Hei-ke) instead of 平氏(Hei-shi).
Similarly there were also many conflicts between 源氏. It’s famous story that 源頼朝 killed most of his brothers and relatives after took the power being afraid of coup.
Last of all we may have to regard another point. 徳川家康 (Tokugawa Ieyasu, the founder of Edo Bakufu) identified himself as 源氏. He thought he is in the lineage of 清和源氏(Seiwa Genji), which is regarded as most influential clan for samurais. Look at the era, it’s far ahead of the origin of Seiwa Genji. That name might have been still influential. Not only him, many people in that era might have concerned it for dignity.
BTW my family name, 渡辺(Watanabe), is 5th or 6th major family name in modern Japan. And it is said that the single origin of this family name is recognized. That is 渡辺綱(Watanabe no Tsuna), who belong to an early branch of 嵯峨源氏(Saga Genji).
So I could say I’m in Minamoto clan just for joking or ice breaking.