The "Language of the Twelve Tribes of Mallus" listed on the Lexi link up there is heavily implied to be Riekspiel, but it's only mentioned in "Soul Wars" and used for spells.

Both. Azyrite is both the languages of scholars and a collection of Common Tongues, as far as we know from it's sporadic mentions. Bear in mind that Azyrheim in many ways is sort of modeled after Victorian England and other Imperial powers of that time. So scholars and nobles use the same language as the common folk.

As far as we know indigenous Aelves weren't a thing until the Aelven Pantheon forced them to be made, largely ignoring the Survivors of the World-That-Was that Sigmar gathered and protrected (which is most likely why even the discontinued armies like Eldritch Council still live in the Cities of Sigmar and Azyrheim, rather than in Lumineth cities). The early Realms were primordial and mythical, dominated by Godbeasts and gods and stranger entities still. Humans, Duardin, and Aelves came to the Mortal Realms sometime before what we know as the Age of Myth, in an era simply known as "Before the Ages", what they did is largely unknown, but those few details we do know suggests they were in hiding, enslaved by evil gods, or had little more than Early Stone Age era tools. These survivors of the World-That-Was did what all mortals do, they had kids, taught them, then died. Over the centuries their cultures would have drastically changed to suit the needs around them. So let's say a Brettonian knight and his estate stumbled into a Realmgate and somehow got to Ghur, they'd have to live nomadic lives to survive in a Realm where crops are hard to grow and everything is an apex predator. Eventually they'd turn to using leather, hide, teeth, and bones instead of metal. Local gods would start to be worshipped. Eventually they become something new. A more specific example is the Khul, the Aridian tribe that Korghos Khul is from, who hail from the Chaos Wastes or Norsca in the World-That-Was, but their nomadic travels into Ghur and Aqshy changed their culture significantly

Azyrheim is actually a city founded well into the history of the Mortal Realms, after Sigmar killed a bunch of gods, forged alliances, and rescued a bunch of human tribes, aelf enclaves, and duardin clans. Some of these clearly had some cultures vaguely reminiscient of their Old World ancestors as an example military orders such as the Myrmidites, Sons of Breton, and Knights of Usirian have been mentioned. But with quite a few changes the Myrmidites and Sons of Breton are both Orders of Demigryff Knights, and the Knights Usirian are caretakers of Freeguild graveyards. They are all three similar to their ancestors, but clearly generations and their environments have shaped them.

So yes. There was a time when survivors with their full memories played a major role, but it's been millenia since then and by the Age of Sigmar their knowledge is as lost to the Free Peoples as much of the knowledge of Mycanean Greece and Ancient Egypt are lost to us. But some bits of knowledge survived. Many folk know there was a world before theirs, names still exist, old gods are still worshipped by small cults, some traditions have survived, and a lot of other tidbits here and there.

Well we know that's definitely not the case for Sigmar as he had to be brought to Azyr by Dracothion, a god that is intricately part of Azyr.

so it's more along the lines of the Incarnates being tuned to the same energies of the Realms, rather than them being manifestations of it. Though Alarielle and Nagash like to pretend their Realms are extensions of them.

For Nagash's part this is disproven by the fact he had to break Shyish twice to get it to kind of be under his control. And Alarielle clearly isn't as intuned with Ghyran as she claims, as large parts of it were taken from her.

The only two gods of the Pantheon of Order that seem to be intracately tied to their Realms are Tyrion and Malerion, and they have a hard time traveling to their counterpart's Realm. Some sources say Teclis is also like this, but that creates a plot hole in how Tyrion and Malerion communicated over the centuries.

Then there's Grimnir and Grungni who spent massive amounts of their time well beyond their own Realms.

Of course there's also Gorkamorka... but Gorkamorka has been stated to be an Elemental Deity and operate by completely different rules than those that his peers operate under.