it actually means that you set a cutoff. when the term you subtract is big enough, you render the integral over the functions finite. in your case I'd say that you have sth like -1/(lambda-A), while A is a very big mass. if lambda is small enough the term won't have any effect on your function...
they interact as they always do. first, quantum field theory should in the limit of low energies always give the classical picture, and second, qft just means that you compute fields and quantizations of these fields are the known particles
well well, maybe I can help you out.
branch-cut singularities are continuous singularities, which means, you have endless singularities from one point on which is the branch-CUT. Now two particles start to exist from p^2=(2m)^2 on. If they have additional energy (see equation for relativistic...