I think these two problems Drisos encountered are not bugs, developers put these on purpose.

* Pay attention the names of the Daimyo's after Hojo: Their names are not "Hojo", they are some other generals taking the control of the Hojo Empire army, and establish the resistance against Mongol Horde. You see, its not "Sengoku Jidai" times, its united Japaneese soil.

* Mongols gets no more reinforcements, because Mongols are quite small tribe, and they never had numerous sources for men on their mainland anyway. On 1230's, there were only 4 million Mongolian people on earth, which invaded China have 100+ million population. So even with hundreds of thousands of Koku, you cant have more Mongol Horsemen. (But it would be nice to have "only Korean" reinforcements)