This thesis is comprised of eight articles in the fields of algebraic, geometric and enumerative combinatorics, as well as algebraic statistics and causality. These works are motivated by problems in the mentioned areas, and have polytopes as their underlying object of study. The investigated properties of these polytopes include the distribu-tional properties of their associated combinatorial generating polynomials, lattice point enumeration, face structure, and properties of their corresponding toric ideals. These investigations, for instance, provide answers to some open questions in combinatorics as well as ne wmethodologies for causal discovery. The main characters are lattice polytopes, simplicial complexes, generating functions, permutations,graphs, posets, and statistical models. These objects often interactin rich and surprising ways.