That is a very complicated question. I will endeavour to answer the bit which I am most familiar with (the empire) and will let others answer the other two categories in greater detail.

On the matter of their language, this is at least because Hunnic may not have been the dominant language of the Hunnic empire. There are certainly other reasons which I will not get into. Suffice to say, there is more evidence to suggest that Gothic (and various other Germanic dialects) was more likely the most spoken language. Many of the Huns took on Gothic names (Attila, for example). (See Maenchen-Helfen, 1973)

On culture, the problem is we don't really know what Hunnic culture was. The fact is more of it may have survived the collapsing state but since we don't really know what it was it's very difficult to trace. Kim (2013) has interesting theories on this matter, some more convincing than others (cf Maenchen-Helfen, 1973 and Thompson, 1996).

As for the empire, the matter of the collapse of the Hunnic state is still not yet settled. Thus this answer will be three fold: first, it will briefly address how the Huns ruled so that one can see how the Hun's political system (at the least) survived the collapsing state. Second, the state did not collapse because of disagreement among its successors but because of Attila's illegal usurpation of the state. His sons actually rallied at the battle to keep the state intact. Third, after their defeat at Nedao, the Huns disappear from history for a time but then reappear under the same model of governance they had prior to the Attila's murdering of Bleda. All of this is to show that, at least in one way, how the Huns empire did survive until they vanish from the historical record.

First, the Huns practiced dual kingship which has been, I think, successfully shown by Kim (2013) to be the case (cf. Maenchen-Helfen, 1973). There is a long tradition of steppe empires ruling with two kings from which the Huns emerged.

When choosing their kings, the Huns appeared to favour the next eldest males within a single ruling family (Mingarelli, 2018; cf. Kim, 2013 who claims all accessions were via strict lateral succession). This is why Octar and Rua both ruled and once the latter chose Bleda as his successor, then Attila came next, etc. Furthermore, the order of wives may have been important (Mingarelli, 2018). It is likely the case that Octar, Rua, Mundzuk (Bleda and Attila's father) and Oebarsius were all from the same mother, just as Attila and Bleda were. We can safely say this because after Attila's death his sons Ellac, Dengizich and Ernak who were all sons of Erecan (Attila's first wife, and we know he had many wives) became the next ruling kings in that order.

In sum, the Huns had two kings. They were chosen from a royal dynastic family that most likely had been established prior to the Huns arrival north of the Black sea in the 370s (existence of the family: Maenchen-Helfen, 1973. cf. Thompson, 1996).

Second, Attila's sons (Ellac, Dengizich and Ernak) did not engage (as far as the evidence tells us) in civil war (Mingarelli, 2018). I will spare you the brunt of the arguments about this but, suffice to say, when Jordanes claims that all the sons tried to parcel out Attila's inheritance, he is implying that every son from every wife is getting an equal share which, as we have seen here briefly, has no historical precedent. Thus it is more likely that a dispute among the two new reiging kings (probably Ellac and Dengizich) was born after Attila's intestate death. There is no hard evidence to say that a civil war broke out before the Battle of Nedao over the inheritance (cf. a particularly vague passage from the chronicler, Prosper).

Therefore, the political system of dual kingship is seen once more asserting itself after Attila's sole rule and just before the western half is lost to the rebelling faction (a discussion all of its own).

Third, we finally can see, after ten years of silence in the historical record, the Huns reemerge under the joint rule of Dengizich and Ernak, both sons of Attila and Erecan and of the very same ruling dynastic family. I don't think I'll dwell here as I think the conclusion is fairly obvious.

This is where what we know turns into what may have come next. Kim (2013) posits that after Dengizich's failed invasion of the Eastern Roman Empire, Ernak who refused to participate because he was fighting the incoming Oghurs to the north, joined with them to form a new coalition state that would later become known as the Bulgars. He argues this for among the names on the list of Bulgarian princes is, none other than, Ernak. But is it the same one? We have no idea.

Given that the Huns naturally had two kings and that one just died leaving an open spot and given that Dengizich had just lost a significant amount of the Huns' field army along with his head, it is certaily possible (assuming this merger occurred soon after) that Ernak might have used the Huns' political structure to resolve the crisis (that is, he invited the invading group to co-rule with him).

So, as we can see the big limiting factor for us to trace what happened to the Huns is the sources available. There are just so few who spoke of them that survive. But, as de la Vaissiere (2015) argued for that group who passed China and India calling themselves the Xiung-nu and the Huna, respectively, the single most impactful idea to emerge from the Huns was just that, their name. Huns were ferocious, political, and intimidating, many an agricultural empire's boogyman.

Pardon the vague citations. Writing this on the bus.