We investigate existence and regularity of a class of semilinear, parametric elliptic PDEs with affine dependence of the principal part of the differential operator on countably many parameters. We establish a priori estimates and analyticity of the parametric solutions. We establish summability results of coefficient sequences of polynomial chaos type expansions of the parametric solutions in terms of tensorized Taylor-, Legendre- and Chebyshev polynomials on the infinite-dimensional parameter domain. We deduce rates of convergence for N term truncated approximations of expansions of the parametric solution. We also deduce spatial regularity of the solution, and establish convergence rates of N-term discretizations of the parametric solutions with respect to these polynomials in parameter space and with respect to a multilevel hierarchy of finite element spaces in the spatial domain of the PDE.