I think the Mongols could have conquered Europe but they definately couldn't have held it. In the first instance the lack of strong central power would have meant the Mongols would either have had to adapt and keep the Barons or exterminate the nobility. Neither is very practical as the first will eventually lead to insurrection and the aecond will need a total re-structuring.
As to pastureland, well yes Europe was very fertile but things like crop rotation and fallow pasture were what kept it that way. It would be impossible to convert farmland into pasture on a sufficiant scale. Remember the general population of Europe is already badly under nourished, hence quite short.
As to actually fighting the Mongols, all the Europeans needed was dicipline and they had dicipline in their infantry, that battle between 26,000 Knights and 20,000 Mongols was just that, add in heavily armoured infantry, crossbows, longbows, those fancy new halbards and it doesn't look quite so good for the Mongols.
What would be needed would be a dedicated Alliance, which is possible given the infidel horde and a Papal edict for a Crusade, and someone smart enough to work out glory charges don't work. Admitedly neither are garrentied
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