We study existence and qualitative properties of solutions to subelliptic problems with Hardy potential and critical nonlinearities on stratified groups. We investigate both the semilinear and the quasilinear case. First, we determine the existence, Lorentz regularity and asymptotic behavior of entire solutions. By convenient transformations, we are naturally lead to study the equation satisfied by the extremal functions to some weighted Sobolev-type inequalities on groups, whose analytic expression is not known. As a byproduct, we derive existence results for the associated Brezis-Nirenberg type problem, depending on the involved parameters. We also obtain non-existence Pohozaev-type results.