Let't say Genghis Khan never unites the Mongols. The Mongol continue being a menace to China and Central Asia, but they never reach the territory of the successor states of Kievan Rus, let alone conquer them. No Mongo empire is ever created, at least non as huge as the one IOTL. ITTL, occassionaly, Turkic speaking peoples raid Kiev and other states, but they are never conquered, at least not for long.
How do these "Russian" states evolve? Do they unite, or do they remain separated? Does one of them conquer the rest?
Structure of the "Kievan" and post-"Kievan" Rus was a little bit more complicated than you are implying.
In the simplest form general schema can be described as following: there was the most senior princedom (Kiev and then Vladimir) and a set of the lesser princedoms which also could have their lesser princedoms. All these princedoms had been rules by the Rurikid princes but, IIRC, there were some occasional non-Rurikids (for example, Lithuanians).
Succession, in general, had been passed by seniority within the whole family with the princes moving to the vacant thrones according to their closeness to the current Grand Prince. IIRC, the schema was already slowly changing toward "father - son" by the XIII: Daniel, Grand Prince of Galitz had been succeed by his son (but it seems that Galitz was somewhat outside of this schema) but after the death of Yaroslav Vseqolodovich, Great Prince of Vladimir, this throne had been given to his brother, while his sons had been given the lesser princedoms.
Consolidation, was happening but there is no reason to expect that it would be happening overnight and most probably it would be around 2 centers: Galitz on the West and Vladimir in the Central Russia with the Southern areas (Kiev, Chernigov, etc.) being up for grabs. Presumably, in OTL the Mongols were instrumental in creating unified Russian state (except that at least half of the "Russian" lands remained under Lithuanian rule) by promoting a single line of the Russian princes so it is up to anybody's imagination to figure out if this would not happen without the Mongolian factor.
How does their societies evolve? Will they still see themselves as members of the same culture, distinct from that of their Western, Eastern, Northern and Southern neighbours? Would their religion and common origin contribute to this sense of shared identity? Would Orthodox religion be an unifying force?
They saw themselves as the members of the same entity before the Mongols, which did not prevent them from fighting and killing each other. Not that the medieval France of Germany had been too different in this respect. Of course, it can be argued that the Orthodox Church got a lot of prestige by providing an "ideological foundation" in the ...er.... "liberation" from the Muslim Golden Horde but for a considerable time the same Orthodox Church was quite subservient to the rulers of the Golden Horde and, based on the fact that the Muslims from the Horde had been gladly accepted to the service of the princes of Moscow, there could be at least some doubts. There was even a vassal "Tsardom of Kasimov" given to the renegade Tatar princes and "descent from the Horde" was as honorable as from Rurik and Gedeminas.
Without the Mongols the Southern neighbors would be Polovtsy and by the time of the conquest there were already numerous princely intermarriages. Eastern neighbors? Probably the Great Bulgar on Volga. Predominantly Muslim but not sure if there were some unusual troubles with them. West and North - mostly usual border fighting which did not change too much after the Mongolian conquest: anyway, they did not come to the Baltic coast. So the main opponents - Lithuania, Poland, Hungary and Livonian "entities". So-called "White Rus" was peripheral to either Kiev or Vladimir so Lithuania may get it anyway and Princedom of Kiev, while still formally prestigious, had been losing its importance by the end of the XII century.
How would "Russian" languge evolve?
Why would there be serious differences in its evolution?
And what about society? Would serfdom eventualy become aswidespread as it would IOTL?
There were centuries between disappearance of the Golden Horde and the form of a "serfdom" you are presumably talking about (started in XVIII century) so who can tell for sure?
Would they be more in touch with western European states (by comparison to OTL), or would religion act as a barrier for stronger links? Would there be autonomous cities, with civil liberties, as in parts of western Europe?
How exactly the Mongols were preventing these links to the "West"?
Would this alt Russia (or alt Russias) still expand southwards and eastwards?
If it ends as a more or less coherent state, it most probably would: important trade routes by Volga and to the East from it, furs from Siberia, etc.