This is actually a really difficult question for several reasons. For one thing, the Dome of the Rock is, itself, actually one of the most significant pieces of evidence that we have about Islamic society at the time. Secondly, and as is perhaps evidenced by the first point, we don't know nearly as much about what was going on "socially, diplomatically, and economically" (and that could be continued as militarily, politically, religiously, etc) as we would like. The 7th century is incredibly "source poor" so we're often relying on accounts written much later which may be historically dubious. I would caution you against anything you find on the internet in terms of "understanding the basics" because a great deal of the material online comes from these religious "traditional" accounts.

One of the longstanding and hotly debated questions that this is a part of in the historiography is "When did the Islamic state come into being?" This is disputed both in terms of the Islamic-ness of the post-Arab-conquest polity, as well as the extent to which or whether that polity should be described as a state.

While that date can plausibly be pushed forwards or backwards, the Dome of the Rock has long been recognized, along with, for example, coinage reforms that happened around the same time, as being strong evidence that by the 691 the Ummayads were in charge of an Islamic state that was both Islamic and a state.

Why is that? Well, for one thing, the construction of large monumental buildings is a pretty good hallmark of an organized state. There are barely any such buildings before Abd al-Malik, who built the Dome of the Rock.

Moreover, the Dome of the Rock is the earliest public religious building yet identified. We have no earlier example of mosque construction. Afterwards, it proliferates.

Moreover, on a religious basis, even though non-Muslim texts identify the Muslims as a separate religious community, the inscriptions on the dome of the Rock and and coins issued during the reign of Abd al-Malik are the earliest examples we have of inscriptions referencing Muhammad. Prior to this, coinage might refer to the "commander of the faithful" in reference to the caliph, but not "Muhammad the Messenger of God".

The problem in terms of your original question is that there's multiple ways to interpret this. Fred Donner, for instance, sees a shift from an early "Community of Believers" conception of Islam to the origins of Islam as we know it. Others have looked at inscriptions within the Dome of the Rock imploring Christians to convert as a shift from an Arab-focused Islam to Islam as universal religion. Others have seen Abd al-Malik's coinage reforms and the Dome of the Rock as part of the competition between the Islamic state and the Byzantine empire. And so on and so forth.

The social aspect of this depends in large part on where you come down on these questions, and what you think of the existence of an Islamic "state" in this time period. It also depends on whether you take the sudden archaeological emergence of coins and monuments as evidence of new state, or go with a more "absence of evidence is not evidence of absence" that takes into account other information that points to the existence of a state, but one that just didn't yet have distinct coinage or see a need to have monuments.

These questions fit into a longer-term debate about identity in the Ummayad period. For instance, it is taken as likely that (by Marshall Hodgson, for instance) the civil war that brings the Ummayad's to power should at least partly be viewed as a dispute about a conquest empire that was now attempting to settle and administer its territory. The Ummayad's practiced one particular form of that, based in large part on family ties in non-Arabian urban power centers (e.g. their capital, Damascus). The Abbasids who overthrew them 60 years later in 750 would do so from a center of power further to the east, with the backing of Arab settlers in Khurasan and the Persian "mawali" converts, along with proto-Shia elements and, based on their own stylings, arguably a greater emphasis on religiosity. That's not to make any strict conclusions about what's going on socially at the time but it's clear that from the time of the conquests and throughout this period this is a society that, whether you're talking about the conquered or the conquerors, is trying to rapidly adapt to a shifting socio-political-religious (etc.) landscape.

Source wise:

On the formation of the Islamic state three articles may be of interest:

• Archaeology and the History of Early Islam: The First Seventy Years by Jeremy Johns
• The Formation of the Islamic State by Fred Donner
• New documentary texts and the early Islamic state by Robert Hoyland

Hoyland and Donner both have recent books out on the conquests, In God's Path (Hoyland) and Muhammad and the Believers (Donner) that may be of interest.

It's old but a classic, but as I mentioned Marshall Hodgson's The Venture of Islam remains a solid and far reaching introduction to the subject.

edit: word edit 2: The earliest coins with "Muhammad is the Messenger of God" were issued during the reign of Abd al-Malik, but not actually by him. Rather, they were struck by a leader rebelling against him. His coinage reforms did eventually however include the inscription as well. See Hoyland's New Documentary Texts.