From a linguistic point of view, we can't really speak about anything verifiably Russian until the advent of the Mongols, which served to split Rus' into what would become 'Russian' and what would become 'Ruthenian' (i.e., Belarusian and Ukrainian). Prior to that, of course, we have quite a lot of texts that some people will call "Old Russian" but that are properly called Old Rusian (one s!) or Old East Slavic. These texts all contain forms that are more or less equally diagnostic of all the modern East Slavic languages (Belarusian, Russian, Ukrainian). After the Mongols come, texts from Northeastern Rus' start to diverge from texts from more westerly Rus'.

It's important to make a distinction between the Rus' and Russia. While Russians might have an excuse for being lazy about making this distinction, we're speaking English and it's easy to be precise with our terminology. In the 11th and 12th centuries, who exactly the Rus' were depended a lot on who was writing the chronicles, but it seems on the basis of the ways they talk about location that many of the chronicle writers located Rus' in the south and west of Old East Slavic speaking lands, and not really in the north or east. Of course, this certainly has something to do with the fact that our earlier chronicle writers were working in southern and western regions. While the arrival of the Mongols might be dated as the 'starting point' for Russian, it's really just when things started to diverge, and it took some time for notable linguistic differences to show up in texts. That's the beginning of Old or sometimes Middle Russian, and for many linguists, the beginning of "Russian" as a useful term we can apply to the past.

In Russian, there's a distinction between drevnerusskij and starorusskij "Ancient/Classical Russian" and "Old Russian", and while they correspond to Old East Slavic and Old Russian in the way they're used, the first term has the pernicious effect of erasing the history and heritage of Belarus and Ukraine and propping up some really nasty linguistic and cultural narratives denigrating non-Russian East Slavs.

So, the short of it is, yes, there were Rus' before the Mongols, and we refer to them as Rus' because it is both not helpful and also harmful to refer to them as Russian.

Musically, Russian culture became a thing very, very late in the day. We must recognise that these people were looking to create a distinct Russian identity by using Russian themes, compositional methods (extended melismata found in Russian folk song, octotonic scales and so on) and conspicuous orientalism. The first Russian composer who I can think of was Anton Rubenstein, but his music was considered to German and not Russian enough. The first real development of a distinctly Russian style of music probably comes from Glinka in his operas. Then we come, in the later nineteenth century to the Five. Rimsky-Korsakov, Borodin, Cui, Mussorgsky and Balakirev. They all made innovative use of folk tunes, folk tales such as Tamara and Sadko as history such as Boris Gudonov. Moreover, they consciously rejected German conservatory teaching, adopting sometimes a deliberately unlearned approach with very odd harmonies and unusual orchestrations. In a way the Five came to rather empathise with the "eastern" element of their character that came from the Mongol conquest. Scheherazade By Rimsky-Korsakov is littered with orientalism. Since the five Russian music took a rather odd path because of the Russian revolution. Composers like Prokofiev, Shostakovitch and Khachaturian, because of the ban on musical formalism, were forced to compose in a Soviet realist manner.