No we don't! Citizenship in the US is not automatic, there is a requirement, "and subject to the jurisdiction, thereof", meaning the US Constitution.<quoted text>
Yes we do. We give US citizenship to EVERY child born in the USA except for the children of foreign diplomats.
Obama was born a citizen of his father's nation, this is a fact you can't deny, and being a citizen of his father's country, he is subject to its jurisdiction, at the moment of birth.
Aliens have never been tax with the responsibilities of the Constitution, if so, they would be citizens the very instant they placed foot on US soil and the naturalization laws wouldn't be necessary.