Cavalry tends to be ineffective in history against a walled settlement, and an army of horse archers seems like it wouldn’t be effective against a group of infantry or archers in an enclosed space. Did the mongols simply recruit mercenaries, did they have their own infantry used in sieges, or is there some other tactic they used? Did the horse archers simply leave their horses and act as infantry?
I answered a question some time ago regarding the Mongol invasion of the Kievan Rus, as well as the Siege of Baghdad (1258). I lay out some details and tactics utilized during sieges. You can find this thread here.
If you feel this doesn't sufficiently cover some of the aspects of your questions, feel free to let me know and I can further clarify on some of the details.