Pyrochlore magnets provide an excellent arena for the realization of various many-body phenomena, such as classical and quantum order-by-disorder, as well as spin-liquid phases described by emergent gauge-field theories. These phenomena arise from competition between different symmetry-breaking magnetic orders. In this paper, we consider a subspace of the most general bilinear nearest-neighbor Hamiltonian on the pyrochlore lattice, parameterized by the local interaction parameter Jz±, where three symmetry-breaking phases converge. We demonstrate that for small values of |Jz±|, a conventional q=0 ordered phase is selected by a thermal order-by-disorder mechanism. For |Jz±| above a certain finite threshold, a spin-nematic phase is stabilized at low temperatures. Instead of the usual Bragg peaks, the spin-nematic phase features lines of high intensity in the spin structure factor. At intermediate temperatures above the low-temperature orders, a rank-2 U(1) classical spin liquid is realized for all Jz±≠0. We fully characterize these phases using classical Monte Carlo simulations and a self-consistent Gaussian approximation.