But how could those countries exist if they wanted to "cleanse" the world of the non-whites that lived there?
Hitler didn't care about Jews and others living abroad. As long as Europe was white, he was happy. And as Italy would be controlling Africa, it was their problem, not Germany's.
For one, he didn't want to annihilate all Jews at first. He wanted to deport them into a distant country like Madagascar, but when Germany started losing the war he agreed to exterminate them.
I don't know how long that would have held up though. I mean Hitler broke his non-agression pact with Stalin fairly easily, and it would be no surprise to me if he ended up invading Italy after he was done with Russia and the UK.
Italy, for it's own ends, wanted an Empire again. That's why they tried to invade Ethiopia. While it wasn't "world domination", they certainly wanted a bigger piece of the world. Japan was much the same. They invaded China in an attempt to grab more land for their own Empire.
That is true and that is one big reason why the Axis failed. They all fought for their own ends, and didn't care a bit about others (Finland, for one, could have eased Germany's war A LOT if it had invaded Leningrad).
Japan would have indeed collided with Germany, and I personally believe it would have resulted in a very similar situation to the Cold War. Former best friends, now worst enemies. Both nations also had nuclear programs, to make things worse.