In the present paper, sufficient conditions for the exponential stability and passivity analysis of nonlinear diffusion partial differential equations (PDEs) with infinite distributed and discrete time-varying delays are derived. Such systems arise in many applications, e.g. in population dynamics and in heat flows. The existing Lyapunov-based results on the stability of diffusion nonlinear PDEs treat either systems with infinite delays or the ones with discrete slowly varying delays (with the delay-derivatives upper bounded by d< 1), where the conditions are delay-independent in the discrete delays. In this paper, we introduce the Lyapunov-based analysis of semilinear diffusion PDEs with fast-varying(without any constraints on the delay-derivative) discrete and infinite distributed delays. Two novel methods are suggested leading to conditions in terms of linear matrix inequalities. The first one provides delay-independent with respect to discrete delays stability criterion via combination of Lyapunov-Krasovskii functionals and of the Halanay inequality. Note that the Halanay inequality is not applicable to the passivity analysis. Therefore, the second method develops the direct Lyapunov-Krasovskii method via the descriptor approach that leads to delay-dependent (in discrete delays) conditions for the exponential stability and passivity. Numerical examples illustrate the efficiency of the methods.