1. It's the politics of the situation. No one at all in the movie is saying Superman shot people. It is never framed that way, and has become such an internet meme with nothing backing it up at this point that I don't get it. Basically, due to Lex's actions, when Supes saved Lois in a politically volitile area, and KGBeast killed all those people just before he got there, it destabilized the region, and caused even worse political actors to storm in and take it out on the villages in the area. Lex used Supes to create a power vacuum.
2. Lex's motives are pretty simple. He hates the very idea of power that is benevolent. His father beat the shit out of him, and no God came in and saved him. He then came to the conclusion that anything all powerful could not be all good, and that is why he wanted Superman exposed. When he sent Supes after Batman he expected one of two scenarios, either Batman killed Superman, problem solved. Or Supes kills Bats, and proves that he is just a fallible as anyone else. As for Doomsday, that was the contingency. The movie hints that he found out about Apokalips at some point, To him, Doomsday is his idea of power personified. It is the Devil, which he clearly believes in more than God. He doesn't care that Doomsday will likely kill him, he is certain that it will kill Supes and prove that he is right.
In other words, Lex can't accept a world where a good guy with the powers of Superman exists. So he sets out to prove that the Man of Steel is not a god at all.