Also, since the US had been part of the British colonies, were early rules about who was or wasn't a natural-born US citizen laxer for those born in Britain and it's colonies than, say, for someone born in France or the Holy Roman Empire?
They were, indeed, eligible to run for the presidency. Article II, Section 1 of the U.S. Constitution includes the following:
No Person except a natural born Citizen, or a Citizen of the United States, at the time of the Adoption of this Constitution, shall be eligible to the Office of President.
Anyone born outside the colonies was "grandfathered in" as long as he was a citizen by the time the Constitution was adopted. There was also a provision that a candidate had to be a resident within the nation for fourteen years, so a new emigrant from Europe, who happened to be a resident at the time of the adoption of the Constitution would fulfill the first requirement, but not the second - until he had lived in the US/colonies for 14 years.