The logo is constructed of trimmed circular paths. The spacing is not equal between the pieces on the same radius. How do I make sure the cuts are made at the right angles?
The cutting angles are automatically right if your shape is a circle, you only split it, remove the unwanted pieces and give the wanted stroke thickness:
In the middle some new anchor points are unserted, the circle is splitted at them using the Scissors tool and the unwanted pieces are deleted.
In the right a new wide stroke has been given.
Object > Path > Outline stroke converts the pieces to fillable areas which can have new strokes:
Warning: Partially deleted circle is difficult to move, scale and rotate, because the original centerpoint is lost. Keep it grouped with a bigger solid circle or square until the placement and other transformations are fixed. Learn how to use the layers panel to manage different parts.
You can also delete nothing. Keep all parts of the splitted circle, only color the unnecessary parts differently or reduce their opacity in the transparency or appearance panel. This way you retain the centerpoint.
Another way is to make the stroke dashed. That way there is no need to delete parts, but only to adjust the gap lengths. The effect will be the same and the angles perfect.
I would use opacity masks on stroked circles to make something like that. Opacity masks are non-destructive and the mask can be easily edited. Masked objects can be duplicated, rescaled, recoloured and their opacity masks likewise edited.
Finish by adding drop shadow effects to each circle.