In this thesis, we are mainly interested in three topics : functional inequalities and their probabilistic aspects, hydrodynamic limits for interacting continuous spin systems and discretizations of stochastic differential equations. This document, in addition to a general introduction (written in French), contains three parts. The first part deals with functional inequalities, especially logarithmic Sobolev inequalities, for canonical ensembles, and with hydrodynamic limits for continuous spin systems. We prove convergence to the hydrodynamic limit for several variants of the Ginzburg--Landau model endowed with Kawasaki dynamics, with quantitative bounds in the number of spins. We also study convergence of the microscopic entropy to its hydrodynamic counterpart. In the second part, we study links between gradient flows in spaces of probability measures and large deviations for sequences of laws of solutions to stochastic differential equations. We show that the large deviations principle is equivalent to the Gamma--Convergence of a sequence of functionals that appear in the gradient flow formulation of the flow of marginals of the laws of the diffusion processes. As an application of this principle, we obtain large deviations from the hydrodynamic limit for two variants of the Ginzburg-Landau model. The third part deals with the discretization of stochastic differential equations. We prove a transport-Entropy inequality for the law of the explicit Euler scheme. This inequality implies bounds on the confidence intervals for quantities of the form E[f(X_T)]. We also study the discretization error for the evaluation of transport coefficients with the Metropolis-Adjusted Langevin algorithm (which is a combination of the explicit Euler scheme and the Metropolis algorithm).