Holomorphic 1/z

I think where you are messing up with the circles excluding singularities is that you neglect the fact that you never need to cross the branch cut. You can evaluate each integral's endpoints with the same angle as the starting points. Only when you actually go around the singularity and return to the same point do you need to specify how you are dealing with the multivalued nature of complex exponentials. For example,

 Spoiler

path 1 doesn't require any special treatment since you end back at the same place while path 2 requires you to tell what level of this pirated-from-wikipedia helix you are on. I'm sure that's why you're getting 2pi's instead of 0's, you are applying branching concept where unnecessary. Closed paths with no singularities don't present the problem of "what level am I on?" because you are always on the same level and end back in the (truly) same place, so forced single-valuedness is not needed.

as for directionality... uh, i did a counterclockwise path. started in the top left corner and went straight down to bottom left corner for the first integral and so on. and yes, expect a sign change when switching clockwise vs. counter. acts like electrical current in that + and - are just definitions used for consistency and don't actually have reasons for being what they are beyond being what dead white guys used. not sure if addressing real issue here...