What if novgorod took moscow's place as the unifier of all russian states? Would it be stronger than OTL muscovite russia? Would it be more connected to the western world? How would being a republic alter european political history?
Moscow as the unifier became possible due to existence of a local princely
dynasty playing a skillful game with the Horde for few generations: getting themselves appointed tax collectors for the Russian Principalities (and enriching themselves in a process), managing execution of the competitors by the Horde, gradually increasing their military strength by absorbing small princedoms, etc. on a later stage strong central power was needed to conduct a multi-century war against the steppe neighbors.
Novgorod, being a merchant republic, had problems with maintaining a steady political course and hardly could have the same degree of support from the Horde: to whom Khan would be giving Yarlik of the Great Prince of Vladimir? The same goes for military strength: the Republic was explicitly avoiding situations in which military leadership (prince of Novgorod) becomes too powerful and routinely relied upon city militia. This was OK for a while but wars with Ivan III demonstrated that these ad hoc formations are easily defeated even by the small detachments of a professional cavalry (mostly Tatars on the Moscow service).
Then, Novgorod was somewhat “peripheral” geographically relative to the Central Russia, which would create the additional problems for your scenario.
Consolidated state as a republic is one more improbability: most of the Russian princedoms did not have a republican tradition and it was one thing to be vassals of the Great Prince of Vladimir and quite another vassal of a bunch of the merchants.
Relations to the West
probably would be better but, OTOH, it does not look like Novgorod managed to get too many novel ideas or technologies from its Hanseatic partners: it did not even adopt their shipbuilding technologies, did not build any ports and completely relied on the imports and exports being completely conducted by the German merchants. Cultural contacts also were seemingly limited: the foreign merchants lived in their own settlement.