How do I create clean line drawings in PS?

If I try to draw connecting shapes with the pen tool, anchor points are deleted, or the shape is closed, and I cannot add to it. 

Could you please post screenshots taken at View > 100% with the pertinent Panels (Toolbar, Layers, Paths, Options Bar, …) visible? 

The Pen Tool works as expected here. 

Instead of stroking paths you may want to consider using Shape Layers with Strokes; just keep in mind that Shape Layers with more than one subPathItems will close them automatically. 

I am having the hardest time executing an idea. 

I want to be able to draw lines and geometric shapes for use in my photos in Photoshop. I would also like to be able to stoke these shapes with a brush, so they look hand-drawn.

 

I have tried the path tool, but my issue here is I am unable to draw anything more complicated than a single shape. If I try to draw connecting shapes with the pen tool, anchor points are deleted, or the shape is closed, and I cannot add to it. 

 

My next idea was to draw the lines and shapes in Illustrator and copy/paste them in. When I do this, I'm unable to stroke the path. 

As what do you paste them? Smart Object, Layers, Path, …? 

And why don’t you apply the Stroke or use a Shape, Pattern or Scatter Brush in Illustrator right away?