Why is Batman so eager to make Superman want to take over the world? He knows how dangerous Superman can be if he wants to be. What makes Superman what he is, is his humanity. Without it, he would be dangerous.
He doesn't, he was trying to make Superman rebel against the government. If Superman did took over the world afterwards (It's left pretty vague) it wouldn't on purpose and the third story would have been Batman vs Dictator Superman.
Batman's problem with Superman, and perhaps all super powered heroes in Miller's version, is that he feels they're not doing enough to help the world with their abilities; the lesson he learned when his parents died is that the world will only make sense when it's forced to make sense. He spelled it out during his fight with Supes in "Returns" that they could've changed the world if Superman hadn't been so goody goody.
Why do the Kandorians have superpowers when they get out of the bottle?
They've never been exposed to yellow sunlight until their escape. It's also possible that, because they were so small, they leeched some power off of Supergirl when she grew back to normal size.