Two novice questions about Sketchup 3D modeling (How to use Solid tool / Find center of circle)

Select move tool and hover over to your circle’s edge, then hover to the circle center, it should snap. Also check this plugin: Guide Tools that places a point at the center of a circle, then you don’t need to hover anything.

Check this for detailed information on using solid tools: Modeling Complex 3D Shapes with the Solid Tools.
Also this plugin might be useful for detecting problems of your groups/components: Solid Inspector².