We investigate the fundamental problem of the nonlinear wave field scattering data corrections in response to a perturbation of initial condition using inverse scattering transform theory. We present a complete theoretical linear perturbation framework to evaluate first-order corrections of the full set of the scattering data within the integrable one-dimensional focusing nonlinear Schrödinger equation (NLSE). The general scattering data portrait reveals nonlinear coherent structures - solitons - playing the key role in the wave field evolution. Applying the developed theory to a classic box-shaped wave field, we solve the derived equations analytically for a single Fourier mode acting as a perturbation to the initial condition, thus, leading to the sensitivity closed-form expressions for basic soliton characteristics, i.e., the amplitude, velocity, phase, and its position. With the appropriate statistical averaging, we model the soliton noise-induced effects resulting in compact relations for standard deviations of soliton parameters. Relying on a concept of a virtual soliton eigenvalue, we derive the probability of a soliton emergence or the opposite due to noise and illustrate these theoretical predictions with direct numerical simulations of the NLSE evolution. The presented framework can be generalized to other integrable systems and wave field patterns.