We've got instances where Grand Duke Mikhail of Vladimir having to remain at Sarai for 2 years from 1313 to 1315, so it got me thinking what entertainment or other sorts of pastimes did it offer residents, also what kind of architecture did it boast, did it have communal places like baths or anything of that sort, what buildings did poeple live in, was there a big market, were there any annual events or other social gatherings, was most of the city common people or was it primarily the residence of the ruling elite, what was the people's ethnic makeup, what kind of places of worship did it offer, were there Buddhist temples as well etc.
We do! Unfortunately I'm not really an expert in this topic, but I'll quote two accounts and leave the links for your own perusal.
The first is by William of Rubruck, a Franciscan monk who traveled all the way to Mongolia between 1253 and 1255. On the way, he comes across the court of Batu Khan, the founder of the Golden Horde. Here's some excerpts from his report:
When I saw the ordu of Baatu, I was astonished, for it seemed like a great city stretched out about his dwelling, with people scattered all about for three or four leagues. And as among the people of Israel, where each one knew in which quarter from the tabernacle he had to pitch his tents, so these know on which side of the ordu they must place themselves when they set down their dwellings. A court is orda in their language, and it means "middle," for it is always in the middle of the people, with the exception, however, that no one places himself right to the south, for in that direction the doors of the court open. But to the right and left they may spread out as they wish, according to the lay of the land, so long as they do not bring the line of tents down right before or behind the court.
Rubruck doesn't seem to have been all too happy with the hospitality of the Mongols:
On the eve of [August 14, 1253] he reached the ordu of Sartach, and the next day the Nestorian priests were dressed in our vestments in the presence of Sartach. As for us, we were taken to another host who was to provide us with lodgings, food and horses, but as we had nothing to give him he did it all meanly. We drove about with Baatu for five weeks, following the Etilia down its course. Sometimes my companion was so hungry that he would say to me, almost with tears in his eyes: "It seems to me I shall never get anything to eat." The market always follows the ordu of Baatu, but it was so far away from us that we could not get there, for from lack of horses we had to travel afoot.
The second account is by that most illustrious traveller Ibn Battuta. In May 1332, he came across the court of Öz Beg Khan, another of the Golden Horde's rulers.
I set up my tent on a low hill thereabouts, fixed my flag in front of the tent, and drew up my horses and waggons behind. Then the mahalla came up - they call it the urdu - and we saw a vast city on the move with its inhabitants, with mosques and bazaars in it, the smoke of the kitchens rising in the air (for they cook while on the march), and horse-drawn waggons transporting the people. On reaching the camping place they took down the tents from the waggons and set them on the ground, for they are light to carry, and so likewise they did with the mosques and shops.
And Ibn Battuta also has some choice words for how the Khan welcomed him:
These Turks do not know the practice of giving hospitable lodging to the visitor or of supplying him with money for his needs. What they do is send him sheep and horses for slaughtering and skins of qumizz [mare's milk], and that is their [manner of] hospitable treatment.
And that's basically it, as far as these two accounts are concerned. I'm afraid I can't really offer much more context, but if you would like to have a more detailed look yourself, there's a pretty long list of the writings of medieval authors on the Mongols available, compiled by one of the most authoritative voices in the field. Hope that helps!