So Scandinavian vikings ruled the Kievan Rus for a long time, is there anyway for this Scandinavian influence to be increased to where it fills the role Byzantine influence had in the development of OTL's east slavic cultures?
Arguably its ancestral culture - proto-Germanic - already did. All West/East Slavic and Nordic cultures share lots of things based on borrowing from that period.
Your mission, should you choose to accept it, is to make *Russia a Nordic country with a POD no earlier than 900AD.
I really hate saying ASB because it stifles ideas on this forum, but this is pretty damn close. The Baltic Scandinavian people have no advantage in population, nor productivity. They have some advantages and some disadvantages in material culture, but Slavic archaeological finds are generally unimpressive even in areas where they clearly dominated, so that's not much indication of anything other than the Slavs being a spartan kind of culture. Some (extremely dubious) arguments could be made regarding an advantage in state organisation and military matters (since the Scandinavians had to learn such things to survive against Western Europeans) but honestly, as soon as that gets transferred over the the Slavic majority, the country becomes Slavic.
Generously speaking, that's exactly what happened with OTL Rus; by the third legendary generation the ruling class is already mixed, and by the fifth, there is a Greek preacher eulogising a ruler with a Slavic name and the title of Khagan, in Bulgarian Slavic, to an audience made of largely Slavs and Steppe Turcs. One generation later, Scandiavians are as much of a foreign mercenary element in Russia as they are in Constantinople, complete with communal violence directed at them.
Even if an ASB were to transport all of Scandinavia's population into OTL's Ukraine in the year 900, they'd be no more than a large minority, and Byzantine culture would be way more attractive to everyone including themselves. This is all OTL - there was a reason Scandinavian royalty sailed to Kiev and Constantinople to take up court positions, some of them quite minor, rather than the reverse. And Russian nobility, including many Novgorodians, did actually do the same: go to the Byzantines as exiles, allies, mercenaries and pilgrims. How many Slavic/Slavicised noblemen went to Sweden or Norway? Can't think of a single one. There were royal marriages and also some Scandinavians who lives in Russia for a time before returning home; that's the best I can do. But none of that means there was a really special relationship by the 12th c.
Seriously, there's only one way this ends in most universes. You know how people say that Portugal conquering Spain means Portugal turns Spanish, and England conquering France means England turns French? Same here.
The Scandinavian presence in Russia today left a mere handful of personal names, almost no placenames at all, no written documents, insignificant linguistic borrowings, nothing at all to suggest they were systemically capable of colonization in the same sense as what happened in England and Normandy (let alone Finland, or worse Faroers or Iceland which I think is the challenge here).
Maybe have the Vikings arrive to Novgorod before the area gets Slavicised, and consolidating the local Chudic/Ingrian tribes into the Principality of Novgorod (which would be called different). The principality gradually expands, incorporating, Karelai, Estonia, and possibly the Arkhangelsk region. It would be called Russia, as it was founded by the Ruotsi, while having no Slavic culture at all. it would orientate itself towards the Baltic Sea, so most probably would end up being Catholic and later Lutheran.
This is only viable if Scandiavians are competitive with the Chud'/Ves'/Karelians demographically, which is a big big big question. The Slavs didn't become competitive outside the core wheat-growing areas around Ilmen until the 14th c. themselves, but maaaaaaybe.
TBH this POD is properly 7th c., not 10th.
Or Novgorod could choose instead to ally with Sweden or Denmark in the face of the Mongols, offering tentative suzerainty in return for military aid. The terrain and other preferred target areas allow this alliance to actually succeed against the Mongols who really have others areas they want more.
This is the closest scenario to sort-of-not-really-Scandinavian-ness.
It's not without problems. Denmark and Sweden were direct and merciless competitors in almost every way to Novgorod, unlike Moscow, where Novgorod invited princes from and where people spoke a closely related language and had the same confession. I think people really underestimate the amount of common feeling among the population of the principalities of North-West Rus. A Novgorodian government that sells out to Sweden likely simply won't survive the next election.
And as you mentioned, Novgorod depended on Zalesye, not the other way. Their diplomats went east, their colonists went east. Even their pirates largely went east.
The only thing that can establish Novgorod as its own centre I can think of is a sort of really early "uniate" church, immediately post-schism, where Novogorod's self-proclaimed Archbishop is fully recognized as such by the Catholics, but permitted to practice a local rite; and this would have to happen during a big civil war in Rus where Novgorod can resist the rest, or immediately post-Mongols.