You aren't far wrong with your approach, certainly for making the crescent shape by using the path finder exclude function.
To round the corners, add two anchor points using the add anchor point tool near to the corner. With the Direct Selection tool, select and delete the corner point. Then with the Pen Tool, add a Bezier curve to join the two open ends.
As far as making everything even is concerned, make sure your initial circle shapes are constrained to a perfect circle by holding down Shift, as you click and drag. Then use the path finder to make the crescent. Having a perfect crescent shape will make the rest of the process much easier.
Once you have made your first perfect crescent shape, and filled it, and have rounded the corners as shown above:
- copy it and paste it
- change the fill colour
- select the crescent, and transform it to rescale it smaller - by
holding down Shift to constrain the proportions.
- Rotate and reposition it.
Repeat the process for the inside crescent.
As for the inner globe just make another perfect circle. However, I don't think the world map looks good. Perhaps find something better, or just leave it out altogether. Simpler logos look better!