Did Steppe cultures have a concept of Europe and, if so, what did some of them call it?
Steppe culture is a bit vague, and could cover anybody from central Europe to east China. From the letters I've seen of the Mongols they tended to adopt the diplomatic stylings of the cultures they conquered. Their understanding of the west was vague, as it constituted nothing more than a series of minor states on the khanate's periphery. Here's a few samples from diplomatic missives.
"I, the Qa'an, the representative of the Heavenly King, to whom he has given power over the Earth, to raise up those who submit to me and cast down those who resist - I wonder why, you, O king of Hungary, when I have now sent envoys to you on thirty occasions, you have sent none of them back to me; nor do you send me in return your own envoys or letter. I am aware you are a wealthy and powerful monarch, that you have under you many soldiers, and that you have sole rule over a great kingdom. Hence it is difficult for you to submit to me of your own volition; and yet it would be better for you, and healthier, were you to submit willingly. I have learnt, moreover, that you keep the Cumans, my slaves, under your protection, and so I order you do not keep them with you any longer and do not have me as an enemy on their account. For it is easier for them to escape than for you, since they are without houses and move about in tents, and so may perhaps be able to escape. But as for you, who dwell in houses and have fortresses and cities - how will you escape my grasp?"
"This is a decree, sent to the great Pope that he may know and pay heed.
After holding counsel with the monarchs under your suzerainty, you have sent us an offer of subordination which we have accepted from the hands of your envoy.
If you should act up to your word, then you, the great Pope, should come in person with the monarchs to pay us homage and we should thereupon instruct you concerning the commands of the Yasak.
Furthermore, you have said it would be well for us to become Christians. You write to me in person about this matter, and have addressed to me a request. This, your request, we cannot understand.
Furthermore, you have written me these words: "You have attacked all the territories of the Magyars and other Christians, at which I am astonished. Tell me, what was their crime?" These, your words, we likewise cannot understand. Chinggis Khan and Ogatai Khakan revealed the commands of Heaven. But those whom you name would not believe the commands of Heaven. Those of whom you speak showed themselves highly presumptuous and slew our envoys. Therefore, in accordance with the commands of the Eternal Heaven, the inhabitants of the aforesaid countries have been slain and annihilated. If not by the command of Heaven, how can anyone slay or conquer out of his own strength?"