Proximal Markov Chain Monte Carlo (MCMC) is a flexible and general Bayesian inference framework for constrained or regularized parametric estimation. The basic idea of Proximal MCMC is to approximate nonsmooth regularization terms via the Moreau-Yosida envelope. Initial Proximal MCMC strategies, however, fixed nuisance and regularization parameters as constants, and relied on the Langevin algorithm for the posterior sampling. We extend Proximal MCMC to a fully Bayesian framework with modeling and data-adaptive estimation of all parameters including regularization parameters. More efficient sampling algorithms such as the Hamiltonian Monte Carlo are employed to scale Proximal MCMC to high-dimensional problems. Our proposed Proximal MCMC offers a versatile and modularized procedure for the inference of constrained and non-smooth problems that is mostly tuning parameter free. We illustrate its utility on various statistical estimation and machine learning tasks.