when a function f(z) is in the form g(z)/(z-z_0)^N, where g(z) is analytic in the vicinity of z_0; clearly there is a pole at z_0, the strength of the pole is simply the coefficient c_{-N} in the Laurent expansion of the function f(z)

For a more sophisticated application of complex integration from advanced physics (quantum mechanics), see this video: https://www.youtube.com/watch?v=0Won5Vs_65E complex integration starts around time stamp 18:15

Use the fact the series can be written as a sequence and apply it to the sequence in the integral expression lim_n->\infty f_n(z) dx of theorem 12.1.

Differentiation has similar meaning as derivatives in 2D euclidean plane with the additional constraint between the slope along x-axis and slope along y-axis prescribed by the CR-conditions. For significance of integration, look up the wikipedia page on "circulation (Fluid Dynamics)": think about a vortex whose circulation intensity you want to find, so you need to compute the circulation integral given in the wiki page, but a vortex has a hole at the center (if you imagine the picture of a vortex), a hole is nothing but a singularity or a point of non - analyticity of the velocity field, so we will need to use Cauchy integral formulas and defamation of contour concepts. Likewise there are many applications of complex integration in electrodynamics but you guys are in a pure mathematics program where the syllabus is not designed to discuss physical application problems. Look up your textbook (Ablowitz book) for applications

14.4) is similar idea

14.3) Before you find the strength of pole, you must find the region where the function f(z) is single-valued, hence you remove the branch cut z=-1 to z=-infinity along negative x axis, the reason you do not select the branch cut from z=-1 to infinity along +ve x-axis is because then it will wash away the pole z=1 and all it neighboring areas and if it does that "strength of pole" will have no meaning because there will not be any z_0 = 1 and its neighborhood. So you must take the branch cut along -ve x-axis. Then simply Taylor expand around z=1 and find the coefficient c_{-1}

Can you be specific with your query? What exactly is not clear about the modulus part?