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Also it is the current law. But Obama's citizenship is controlled by the law in effect at the time he was born, not the current law.

Which would have been the Immigration and Nationality Act of 1952, also known as the McCarran-Walter Act. The text of that act is available Here and according to that even if Obama had been born outside the U.S., he would still be a natural born citizen through his mother.

Maybe, but maybe not. Obama's mama was not yet 19 when he was born. At a minimum, if Obama was not born in the U.S. (and I'll admit that is a big if), his mother should have been required to prove she continuously resided in the U.S. for the requisite 5 years required by this law:

From Sec 1401 -- (g) of the doc you linked:
a person born outside the geographical limits of the United States and its outlying possessions of parents one of whom is an alien, and the other a citizen of the United States who, prior to the birth of such person, was physically present in the United States or its outlying possessions for a period or periods totaling not less than five years, at least two of which were after attaining the age of fourteen years: ... This proviso shall be applicable to persons born on or after December 24, 1952, to the same extent as if it had become effective in its present form on that date;