Perhaps I can explain. The sith have been existing in secret for a thousand years following their alleged demise at the hands of the Jedi Knights. But only two existed at any given time-a master and his apprentice. Once the master died his student became the master and chose his own apprentice. This went on for a thousand years to the time of Darth Sidious, who was trained by his own master and then took Darth Maul as his sith apprentice.(Some of this might bgorder on EU so I aplogise if it strays abit outside the accepted lines)
Tatooine is TECHNICALLY in the Republic, hence Padme being surprised at the slavery, but to all intents and purposes it was not.
QGJ's absolute confidence that Anakin would have been detected quickly has always convinced me that medicial testing for infants is standard procedure throughout the Republic. A mere few thousand Jedi, busy with other duties, can not go around testing trillions of people.