Photoshop :: Mysterious Paths In Pen Tool Appearing?

I'm a keen amateur on the computer and run my own construction company. We currently have a steel ruler as our logo, but I'm trying to get a spiral staircase also. Not sure if two logo's is allowed; the steel ruler has our company name imprinted on it and I thought it was cool way to display our name. Probably severley faultered from a pro.
 
Was hoping to find some vector art of a helical staircase but not much out there. So we made a couple of beautiful helical stairs for a client, and I'm trying to trace that and use it as our logo also.
 
I'm tracing it in vector art because we have a Roland GX24 vinyl cutter and I'm making my own banners etc.
 
I have three queries:
 
1:      The pen is showing up some strange invisible paths which only appear in the plugin cutstudio (where I haven't closed the paths off I presume?). How can I get the paths to be as drawn?;
 
2:      When I go to cut this, the vinyl cutter will treat the paths as single slices. This design will be as if drawn, so I will want a way of creating two paths to the strokes, probably 2mm width apart. That will then leave a piece of vinyl 2mm in width so when I peel off, the vinyl will stay there; like a pen stroke of a handwritten drawing.
 
3:     What snapping should I be using? I've always had problems with points or paths not being snapped together perfectly. The cutter will end up cutting some of the other adjoining 2mm vinyl and after time this will cause the vinyl to peel off.