He was the first president of China. So I'm not sure what you mean by the first part of your question.

The reason why the first president of the "Beiyang" government was Yuan Shikai and not Sun Yat-Sen was because Yuan Shikai was the premier general of the Qing Dynasty, with the strongest military force at his command. The Republican government made a deal with Yuan that would allow him to be the first president if he was able to get the Qing Emperor to abdicate, thus resulting in a (relatively) smooth transition of power to the government.

Both Chiang and Mao claimed to embody Sun's ideas, principally his "Three Principles of the People," which were "Nationalism, Democracy, and Social Welfare." However, the two differed sharply on what "Democracy" and "Social Welfare" meant. Chiang interpreted Democracy as being more or less some sort of republican central government, while Mao interpreted as it being more of a government for the people (i.e. communism). On the other hand, Chiang shied away from the "socialist" implications of Social Welfare, instead choosing to understand it as a form of economic system where a centralized government would provide a foundation by which to take care of the people and restrain capitalism, while the Communists took it to interpret that Sun supported social welfare, and by extension, socialism.

To an extent, Sun was "the" Chinese Founding Father. And both Chinese government contenders claimed to be the embodiment of his ideas, much like today most American politicians, for example, attempt to compare themselves to various American Founding Fathers, like George Washington and Thomas Jefferson.