•  Convert the solid shapes into paths/outlines (full drawing at once)?

dimitri,

Clearly, the drawing was originally made with stroked paths, and then someone performed the destructive act of Outline Stroke, so what you wish was literally destroyed in the version published, as a common simplification.

So unless you can get the original, you are stuck with other (non) solutions.

But you can still fulfil the real purpose "so that I can easily play around with the thickness of the outline":

How about just applying Effect>Path>Offset Path, trying out different offset values wtih Preview ticked, and maybe using Object>Expand Appearance to get new real accurate paths?

Here is a loosely drawn simple sample (a Compound Path and a simple path, ungrouped), as Effect to the left and Expanded to the right, starting with a path width of 20pt (in the middle) and using an offset of 5pt (at the top) and an offset of -5pt (at the bottom).

Click to get closer

You can try Trace Image in Ai and then create outlines from strokes, that will allow to change them.