I am not an expert on all of these questions, but I’ll try and give you a quick run-down of some of them.

The Catholic church (15th-16th century) said that first man to see new land would get a reward? Why would the Church want this? What are their interests?

Taking possession of territory was much more complicated than just whoever saw it first got it. There was a lengthy and complicated history of legal possession, which was rooted in Roman and Medieval law. To generalize, the Church acted as an arbitrator between Christian powers to avoid international conflict, directing that conflict instead at those who were not Christian. The Pope granted European powers the right to seize territory of non-Christians. This was originally directed at known non-Christians in the European sphere, especially rising Muslim rivals to the east. This was later used to justify involvement in islands in the Atlantic. Later, when Columbus arrived in the New World, he immediately carried out a series of solemn ceremonies through which he demonstrated that he too was following the accepted international (well...European) law of that time. Shortly thereafter, the Pope granted Spain the exclusive right to govern most of the New World, provided that Spain christianize the population. Later, the right was clarified to extend only to non-Christians who carried out acts of war against Christians or members of the clergy. This in turn led to the development of the definition of a “just war” which led to the first inklings of international law.

Another legal precedent that existed largely outside of Church oversight was the idea of taking possession of empty land. Again, this took various forms over the hundreds of years that Europeans sought to exert control and build empires. In some cases, possession was recognized because of Roman and Medieval law. If land was unoccupied, it could be taken by a European power as long as they declared their intention to occupy it a la Roman law. Later, English, French, and Dutch colonizers extended this precedent beyond mere intention; one needed to physically occupy and develop the land to take possession. Thus, taking possession had specific benefits for the Pope and European powers, especially spreading the faith and acquiring resources and labor.

Schools are teaching us the wrong story about Columbus? He was sailing to India to find gold? But he saw this new land (he knew this wasn't India) and stole the gold native americans had and made them slaves?

I don’t know how you were taught about Columbus in school. In short, Columbus sailed west to reach the Indies. He unexpectedly ran into the Americas. Historians still debate whether or not he ever knew he was not in Asia. Most likely, he never realized or refused at the very least to believe that he had failed in his endeavor. In any case, he travelled back and forth four times to the New World; he was also the governor there (though few would say that he was an effective administrator). On his first voyage, he took slaves to be used as translators and to Christianize them. During his governorship, Columbus and other Spaniards took slaves who resisted Spanish efforts to spread the Gospel. Keep in mind that at the time, slavery was completely legal and officially sanctioned to be taken if they were acquired in a “just war,” which the Pope defined as aggression to proselytizing expeditions.

Is this the start of slavery? The Africans had a more advanced civilizations (economically, politically) that Europeans back then, before they became slaves?

Slavery had existed for millennia and took many different forms in many different cultures around the world. So no, it was not the start of slavery. I also don’t really understand the second part of this question. African societies were incredibly diverse, so it is really impossible to summarize them all. Additionally, the terms “advanced” and “civilization” are troubling as they both have certain cultural connotations that historians are working to move away from whenever possible.

Again I’m generalizing: Slavery in the New World grew out of past international traditions of slavery, various existing labor systems utilized by both Europe and indigenous societies, new economic models that emphasized the extraction of resources and later the production of cash crops, and the availability of groups who could be compelled to work. They first used indigenous people as slaves, following a model of encomiendas which already had been used in the Old World. Indigenous slavery, as previously mentioned, was justified because the indigenous people resisted efforts to spread the Gospel, though this was hotly debated (the legal justifications for slavery changed over time). Unfortunately, indigenous Americans had a tendency, especially early on, to die in great numbers of disease. To fill the void, Europeans turned to African slaves. This developed into an unprecedented system of chattel slavery that brought millions of Africans to the New World against their will.

Mayflower was shipping dutch and english separtists, also black slaves in US?

To my knowledge, there were no people of African descent aboard the Mayflower.