In this paper we provide a comprehensive approach to the spectral fractional heat equation that combines purely analytic and probabilistic perspectives. Furthermore, we give two new results on the monotonicity properties of the spectral fractional heat diffusion with respect to the fractional parameter. The first result deals with the spectral fractional heat kernel, evaluated at the initial singularity. The second result considers the probability for the corresponding stochastic process of being confined in a subregion of the domain. In both results, the monotonicity property depends on the size of the first non-zero eigenvalue. The case of homogeneous Dirichlet boundary conditions are addressed in detail here. Neumann boundary conditions will be taken into account in the companion paper [Serena Dipierro, Giovanni Giacomin, and Enrico Valdinoci, Diffusive processes modeled on the spectral fractional Laplacian with Neumann boundary conditions, Springer INdAM Series].