Editing Shapes and shape paths

Grabbing a vertex and moving the whole shape happens when all of the vertices are selected. If only one vertex is selected, then it just moves that vertex. Clicking on the Path property of a shape layer in the timeline panel will select all of the vertices. You can drag a selection around one vertex (or any number of vertices) when it's in that mode to select a specific one (or specific set).

Double-clicking a vertex will launch the bounding box that lets you Transform the shape.

If you want bezier handles on a shape layer that has been created with one of the parametric tools (Rectangle, Polystar, etc.) you have to right-click on the shape (Rectangle 1) in the timeline and select Convert to Bezier Path.

When you want to edit a Bezier path the easiest thing to do is select the pen tool (g) and then use the Alt/Option and Crtl/Cmnd keys to convert the pen tool to selection and convert point tools. Select the path you want to edit in the timeline, select the Pen tool, hold down Ctrl/Cmnd and then observe the cursor as you move over the path. When you get a pointer with a white fill you can drag a selection around one or more points to select them then release the Ctrl/Cmnd key and move the points. Holding down the Alt/Option key will let you adjust the Bezier handles with the Convert Vertex tool.

I explain it all in this short video:

I'm not clear what you're getting at. You simply click outside the object to deselect. the rest is determined by where the various shape entities reside hierarchically, so it may help to actually revela the structures in the timeline rather than relying on hit-based selections in the composition.

Mylenium