I've thought about this a lot too, in being a Trekkie and, among other things, an Ethics and Politics student.

From what I can see, the main cause of extremism and the refugee crisis is successive war, and the subsequent destabilisation and power vacuum left from such war which has led to poverty, and people fleeing the conflict for a better life as it's the only viable option.

The federation would most likely do what the US, UK and other Western nations are attempting and somewhat succeeding in, which is peace in the region. Like the recently established ceasefire in Syria. What the federation would most likely do differently however is the handling of the refugee crisis, as the federation is based on the idea of, as you said, a united humanity, they would redistribute wealth and resources to accommodate and help the people. The cost and financial risk would be irrelevant to them, as they'd view the globe as one without national borders and would use the earth's resources for the common good of all.

With such global investment and a greater redistribution of basic needs (education, education, education) from a united world government, the motivation to flee your home or join a terrorist organisation would most likely be eliminated. If you had what you needed, you'd see the prospect of relocating as incoherent. And because of this, such people would have less anger and discontent to manifest into terrorism, and Islamic fundamentalism (like Nazism) would lose its main source of recruitment and momentum.

Overall, I think that if we had a federation with a united Earth within it, the root causes of the extremism and a refugee crisis would not have occurred, as Earth would most likely be in a united position, using the richness of the Earth to provide for all.

That's unless the Borg come back and kill us all.