"How did the dinosaurs get from the island to the mainland?" They swam, built or commandeered boats.

"How did they usurp the South Americans?" The assimilation was very gradual, not dramatic.

"Why did they start building robots? And how?" They needed human-like intermediaries to help them deal with humans. The technology for robots is well advanced by this time. Robots are being built by many developed countries.

"Why did the rest of the world's population let them take over South America?" The takeover was transparent enough that by the time the rest of the world understood what was happening, it was too late. Since they control all of South America (and the weapons of those countries) the other nations fear their strength. There is also a sentiment among some parts of the human populace to leave the dinosaurs alone, "let them have South America, as long as they don't bother us.