Fixed point theory in generalized metric-type spaces is useful for studying nonlinear problems where classical metric assumptions are too restrictive. The main contribution of this work is to establish new fixed point theorems for interpolative contraction mappings in strong extended $s$-suprametric spaces. These spaces are generalized distance frameworks that extend several known suprametric-type structures while preserving conditions suitable for convergence analysis. Interpolative contractions involve mixed-distance contractive conditions and therefore apply to mappings that may not satisfy the classical Banach contraction principle. Examples are included to demonstrate the efficacy of our findings. The existence of solutions to three categories of nonlinear problems, third-order boundary value problems, Fredholm integral equations, and dynamic programming problems, were demonstrated in the applications of the developed fixed point theory.