This kind of question involves rather subjective and relative terms, like positive and negative legacies, which are bound to trigger a "yes and no" answer.
Not only these terms are relative in themselves, they are also relative to the historical actors — meaning "adversaries".
There is the famous Monty Python's "Life of Bryan" moment What have the Romans ever done for us? where this problem is perfectly illustrated. The Romans are seen as the empire that by definition, the easy or the hard way, is supposed to have always brought civilization, (although the Pythons are overdoing it by suggesting the Romans had brought wine to the Middle East!). — In the end, anyway, the Romans all but demolished Jerusalem, evaporated any remnants of Jewish polity and reduced the Jews to a diaspora for a millennium and a half. (And if the Jews survived as a culture, a language and as a people it was not because of anything the Romans had brought, but because of the things the Israelite had themselves invented and inherited from the much older civilizations closer to them.)
The civilizing impact of the Roman conquest is what the European colonialists always assumed of themselves, of course. Within that picture the anti-model were the invading peoples, from the Huns to the Mongols. That perspective can and must be relativized, even if only up to a point.
It is thus not unfair to look for a "positive Mongol legacy" (although it sounds like a refurbishment of the Python sketch).
The immediate problem with such a legacy is that, once envisioned, it's too easy to find! — in the Nietzschean form of "what doesn't kill me makes me strong". Of course an invading and destructive attacker is bound to trigger some residual "positive" response from a surviving victim and to bring forward and encourage some potential. It's a Darwinian scenario where the surviving animal has at least confirmed as "positive" those few assets it has which allowed it to keep breathing, albeit mauled and bloody.
For example, the Mongol pressure had as a positive effect on the Hungarian kingdom the development of more and bigger fortified castles, and up to a point it might have contributed to the strengthening of the royal power (hence of the state) against the nobles (although that power remained shaky). In all Eastern Europe it is the Mongols that triggered the special development of the mounted archery and light cavalry - which ended up as the Hungarian Hussars and Polish Uhlans as a response and as a borrowing from the steppe cavalry. (Wallachian and Moldavian armies favored the bow and the light cavalry Mongol-style, and that might have contributed to their initial successes against the Ottoman, especially that volens-nolens they had became experts of scorched-earth strategy - while some "points" inherited from Mongols and Turks became a local specialty, like in the case of the impalement proclivity and overall politics of Vlad Țepes-Dracula, who Ivan the Terrible might have emulated, according to some; see this; same idea here and here).
But there are also more direct and less ironic "positive results" brought by the Mongols, and they can be described as increasing in significance on a scale at the top of which we can situate Russia-proper, that is Muscovy, although the relativity and the contradictory aspects of those positive things cannot be eliminated.
On the lowest part of the scale we can put such effects as those mentioned above: Mongol pressure indirectly triggered military development (of fortifications and cavalry) and the centralization of some states.
This effect of centralization represents a still indirect result, but more explicitly positive, when it leads to the development and independence of some states as a result of a change brought by the Mongols to the regional balance of power. — Muscovy fits this model more than others. Wallachia possibly owes its initial independence from both Bulgarian tsardom and the Hungarian kingdom to the new balance of power brought by the Golden Horde (Crimean Tartars). Moldavia, created as a Hungarian buffer against the Mongols was able to secure its relative independence under a similar logic. The Cossack Hetmanate developed also as a buffer state between stronger powers, on the same trends and in the same region.
But, more to the north, what happened to Muscovy goes much farther in that direction. Initially it acted as the main instrument in the Mongol domination in the area, then became the strongest Mongol vassal and started absorbing the other principalities of the north under the Mongol tutelage. After a while though, it not only became independent from the Mongols, but destroyed much of their power and in the end became the geopolitical successor of the great Mongol empire, between Poland and Japan.
That is the main direct heritage that the Russian empire received from the Mongol one: a very specific Eurasian geopolitical position, which comes with very specific constraints and opportunities. Whether that counts as a "positive" asset cannot be decided in absolute terms, and that is even less the case for the political model that this asset encouraged: autocracy. Beside the initial Roman-byzantine imperial model, the Muscovite and Russian autocracy seems strengthened by some absolutist Mongol features that have possibly ambiguously been contributing to the Russian political ethos. It is of course debatable and obscure to what extent it is the geopolitical context that determines the political model, and to what extent, then, the "Mongol heritage" is "Mongol" at all, or it is simply determined by that context, as a more generic factor.
At a lower level of that scale, it is like saying that a virus that doesn't kill you gives you immunity. In Muscovy's case this went much farther: it amounts to one's capacity of surpassing ones master, to getting an imperial inheritance, where Russia in a way became the "new" Mongol empire (geopolitically), much more than it became the "new" Byzantine one ("the new Rome", that the new imperial power pretended to reincarnate after it conquered Kiev and after its name became "Russia").
Aesthetically that political heritage is clearly reflected in the design of the initial crown of imperial Russia, the Monomakh's cap:
In this sense even the most positive Mongol heritage in later Russia is a political system and a geopolitical situation that are not without their contradictions.
As for more direct and arguably morally or aesthetically positive effects or borrowings like those mentioned by the OP (opening of the commercial routes to the east, cultural influences like music - by which the OP must mean folk music) they can easily be dismissed. The Eurasian commercial routes that the Mongols controlled were far from Muscovy, too much to the south. Maybe if entire Europe had been conquered by the Mongols we would have ended up with a unified Eurasia from France to China with a flourishing commerce (but maybe we are lucky we'll never know for sure!). Thus, Eastern Europe remained a periphery of the Mongol realm, and the Chinese technical and political advances that the Mongols adopted (bureaucracy, paper, gunpowder) didn't radiate this far. Chinese gunpowder siege weaponry was made felt in Hungary at some point, but not in a humanly progressive sense. Last but not least — as far as flourishing commerce is concerned —, Mongols continued the slave trade north-to-south (from the context of which the very word slave etymologically derives,
from "Slav"), in the end through Italian mediation of the Crimean ports.
As for the typical Russian music, it is very clearly closer to the rest of eastern European folk music, from Poland to the Balkans, one that is different from (and in my opinion superior to) the folk music of western Europe, but has nothing to do with the Mongols.
