Much of the Mongol empire functioned more like a patronage system than an empire that directly ruled over a territory (e.g., Rome). In the case of Russia, after the initial invasion and conquest, the Mongol leaders installed princes they could count on to be loyal and provide tribute. (Interesting side note: the reason Moscow gained the prominence it has to this day is because the prince of Moscow told the Mongol khan that the most powerful Russian prince at the time (who was prince of Tver) had murdered his Mongolian wife, an allegation that was probably false but that caused the Mongol army to ransack Tver and kill its prince while promoting the Moscow prince as a reward. The city of Tver is still bitter about this.) Given that the Mongols were a nomadic group, such a system of ruling better suited them than a more sedentary and established approach, like the one used by Rome. And although it means that there aren't the same types of physical legacies of the Mongol Empire across most of its former expanse, there are still remnants that can be seen in language and in legal codes. For example, the phrase in Russian for "Let's go!" (Davai) is basically a transliteration of the Mongolian.

What do you mean by this? The Mongol Empire and its successor states did sponsor much notable architecture.