The world will always have conflicting interests.
That's true, but in a world of democratic governments those interests don't have to resort to violence. They don't even necessarily have to get what they want as long as they believe they can get it without bloodshed.
We are most likely going to have an economic breakdown soon, followed by installing of extremist governments, thus leading to war.
The economic breakdown I grant you, but it won't lead to extremist governments, it will lead to a long-needed recognition that our current financial system is unsustainable and trying to do something we can no longer really do (that is, get larger and larger). The resulting reconstruction will be a positive influence, not a negative one. We might see some war because of it, but ultimately the war trend will continue as it has been-downward.
Living in harmony is basically impossible. How do you balance the needs of hundreds of different powers?
Living in harmony is not impossible. It just requires an honest, open political system and a culture of peace. These things are both growing and spreading. The Western world is pretty much there in regards to one another, for example.
Actually, I'll disagree on that second point for three reasons, at least two of which play together:
1. It relies on politicians, people working in a field where egotism and self-aggrandisement are survival traits, to be readily willing to admit when they are incorrect.
2. It relies on people having a great comprehension of economic theory, a field where even economists are divided
3. It relies on people being unwilling to accept quick and easy, but radical nostrums promised by parties that themselves live on one of the far ends of the political/economic spectrum.
We've already seen a sharp increase in radicalism in Greece, courtesy of Golden Dawn and even Syriza. Historically, economic or political unrest has almost always resulted in a sharp increase in extremism, whether it is in Greece, Germany, Russia, Hungary, medieval Europe, or any of another host of examples. It may not result in an extremist government being ushered into power this time, if we're lucky, but that is far from a guarantee.
The western world, yeah sure. But there are hundreds of different cultures, races, ideals and such that absolutely hate each other. How will you get rid of that hate?
The same way we did it in the West. Slowly and gradually deepening the bonds between people through communication, trust, trade, and mutual development. It'll take time, there will be growing pains, and I'm not saying that trade and communication will be a simple heal-all panacea (fun fact: Nazi Germany's biggest trade partner in 1939 was France, and their biggest trade partner in 1942 was the USSR). However, I believe that it can be done in the rest of the world as hatreds become rivalries or even partnerships, as between France and Germany, Poland and Germany, Austria and Germany, Denmark and Germany, Sweden and Denmark, the Netherlands and Spain, Hungary and Austria, Italy and Austria, Italy and Yugoslavia/Serbia/Croatia, Italy and the Papacy, Italy and Italy...