Rotate around any vertex

Lock the object to be rotated first.

Turn on Vertex snap.

Select your axis to rotate on.

Hover over a vertex.

Rotate.

Note – you can also access snap with shift right click

Creating a spline on a sphere

Use snap to vertex or face, etc.

Start with all corner vertices.

You can make the line wrap the sphere by converting vertices to bezier.

Note – you might need right click to reset tangents.

 

Place a sphere on a vertex of a larger object

Base to pivot, auto grid, snap to vertex

[Omit vertex snap if you want to just place the small sphere anywhere on the larger object]