There are a bunch of questions here, so I will just do my best to answer them as efficiently as possible.

First of all, one can integrate a propagator to a force law. A massive propagator $1/(p^2+m^2)$ induces a force law $e^{-rm}$, in natural units, while a propagator $1/p^2$ induces a force law of Coulomb type, eg. $1/r^2$ in 3+1D. Notice that the first has no small r divergence while the second does. This is the basic statement, though it is modified in many interesting ways beyond tree level.

In formulating scattering problems in gauge theory, one must be quite careful about how to specify boundary conditions of the gauge field at infinity. Peskin and Schroeder have a great description of this, but there are also some new perspectives coming from Andy Strominger and his collaborators. See, for example, these lecture notes (there are also videos on youtube).

Hope this helps!