why do we operate with graphical models in VAE, if there are no probabilites involved?
Not sure which type of code you were looking at but e.g. here they do sampling of random variables. The encoder deterministically maps the input X to mean and standard deviation vectors. Using these vectors you sample Z from factorizing Gaussians with these means and standard deviations. The samples of Z are the again mapped deterministically back to X by the decoder. The reparametrization trick is used to propagate the gradients through the network while only having to sample from standard Gaussians.