Quantitative aspects of Hamiltonian Floer theory have been proven useful in studying Hamiltonian dynamics. In recent years, cohomological operations with characteristic p coefficients have also generated surprising results of Hamiltonian diffeomorphisms. However, most applications only worked for a rather restrictive class of compact symplectic manifolds. Continuing our work on the integral Arnold conjecture, Guangbo Xu and I are building a comprehensive package of integral (and mod p) Hamiltonian Floer theory using FOP perturbations. I will discuss the foundational aspects and indicate how these tools lead to applications (also joint with Shelukhin and Wilkins) in Hamiltonian dynamics on general compact symplectic manifolds, including various sufficient conditions leading to infinitude of Hamiltonian periodic orbits, like absence of abundance of genus 0 J-curves, existence of symplectic degenerate maxima, or existence of non-contractible orbits.