  Gapless nodal quasiparticles emerge at a low-energy regime of high-$T_c$
cuprate superconductors due to the $d_{x^2 - y^2}$ gap symmetry. We study the
unusual renormalizations of the Fermi velocity $v_F$ and gap velocity
$v_{\Delta}$ of these quasiparticles close to various quantum critical points
in a superconducting dome. Special attention is paid to the behavior of the
velocity ratio, $v_{\Delta}/v_F$, since it determines a number of observable
quantities. We perform a renormalization-group analysis and show that the
velocity ratio may vanish, approach unity, or diverge at different quantum
critical points. The corresponding superfluid densities and critical
temperatures are suppressed, slightly increased, or significantly enhanced. The
effects of three types of static disorders, namely, random mass, random gauge
potential, and random chemical potential, on the stability of the system are
also addressed. An analogous analysis reveals that both random mass and random
gauge potential are irrelevant. This implies that these fixed points of the
velocity ratio are stable, and hence observable effects ignited by them are
unchanged. However, the random chemical potential is marginal. As a result,
these fixed points are broken, and thus, the instabilities of quantum phase
transitions are triggered.
