Here are the remaining answers since I very much doubt anyone else will be giving it a try. Feel free to discuss if you disagree.... Thanks for all who'd participated.
1) The Mongols overran Xixia and also the Jin empire in North China relatively quickly. However, they would take about a generation to conquer the last of the three feuding kingdoms within China proper, the Southern Song. Why and in the end, how?
One word - geography. Southern China is a land of rivers and paddy-fields, and uniquely unsuited to the main Mongol style of warfare, mainly large cavalry formations. They have no navy to speak of, to skirmish on the Yangzi.
E.g. the Mongols spent 10 yrs or more beseiging the city of Xiangyang. Surrounded the city completely but the Song were able to cont supplying the defenders via the Yangzi river. In the end, it took a mutiny of the Song navy before the Mongols could make headway.
The Mongols also had trouble with the better technical resources at the Chinese' disposal like ancient rockets, explosives etc. But they learned to adopt them.
4) When, how and why Buddhism took root in China, the only foreign religion to pervasively permeat all levels of the Chinese polity?
Someone had answered the timing correctly. The reason why a foreign religion was able to take root at this peculiar time was that for the Chinese, this was a time of upheavel and chaos. The Han empire had collapsed and in its demise, barbarian tribes poured in fr the north and filled N China while the south was divided betw a few kingdoms. The Chinese seeked solace in religion and Buddhism arrived in time to fill this need. Daoism too thrived and developed much during this period.
5) Why was the Tang so powerful militarily (at least in the beginning)?
The Tang emperors viewed themselves as cosmopolitans. They seeked to be the Chinese Son of Heaven, as well as being the Khan, Turkish-style. The Tang imperial house and its immediate followers were half-Turkish themselves. Hence they were more open-minded and receptive and more martial, than normal typical Chinese rulers.
In particular, they were not averse to signing up the nomadic tribes as their vassals (which was something traditional Chinese rulers could not envision) in the barbarian style. The Tang also took in talented foreigners to command their armies. Like the Korean general who led the Tang army to defeat in the Battle of the Talas River against the Arabs (despite breaking up the Tibetans earlier whom the Chinese feared would join the Arabs in alliance). Ironically, the Chinese would get on very well with the Arabs later on.
Or the Sogdian An Lushan who commanded 200000 troops at the frontier, who would rebel later and weaken the Tang permanently. The Tang also took up the nomadic style of warfare. Tang light horse conquered all the way to the west.
8) How did the Silk Road begin?
It began when the Han emperor, Wudi, sent an emmisary to the Yueh-chih in the west, to try to make an alliance against the powerful Xiongnu confederation to the north. The Yueh-chih had been driven to the west by the Xiongnu and had a grudge. Unfortunately the Yueh-chih refused and went on to conquer a large swath of Central Asia and N India, becoming the Kushans.
Despite this, the emmisary managed to return to China after many yrs and brought back news of other nations in the west. The rest is all history.
10) Why did the Sui collapse?
The second emperor embroiled the empire in too many escapades which it could not afford. He tried to conquer Korea, but failed despite five campaigns and mobilization of up to a million men. He pushed for the completion of the Grand Canal, which took a heavy toll on the local populace (despite being a sound construction in the long run).
In the end, widespread unrest broke out and his commanders and officials rebelled and began a struggle for power, particularly the house of Li (who would be the future Tang dynasty).
