  We investigate the stability and $e^+e^-$ pair creation of supercritically
charged superheavy nuclei, $ud$QM nuggets, strangelets, and strangeon nuggets
based on Thomas-Fermi approximation. The model parameters are fixed by
reproducing their masses and charge properties reported in earlier
publications. It is found that $ud$QM nuggets, strangelets, and strangeon
nuggets may be more stable than ${}^{56}$Fe at $A\gtrsim 315$, $5\times10^4$,
and $1.2\times10^8$, respectively. For those stable against neutron emission,
the most massive superheavy element has a baryon number $\sim$965, while $ud$QM
nuggets, strangelets, and strangeon nuggets need to have baryon numbers larger
than $39$, 433, and $2.7\times10^5$. The $e^+e^-$ pair creation will inevitably
start for superheavy nuclei with charge numbers $Z\geq177$, $ud$QM nuggets with
$Z\geq163$, strangelets with $Z\geq 192$, and strangeon nuggets with $Z\geq
212$. A universal relation $Q/R_e = \left(m_e - \bar{\mu}_e\right)/\alpha$ is
obtained at a given electron chemical potential $\bar{\mu}_e$, where $Q$ is the
total charge and $R_e$ the radius of electron cloud. This predicts the maximum
charge number by taking $\bar{\mu}_e=-m_e$. For supercritically charged objects
with $\bar{\mu}_e<-m_e$, the decay rate for $e^+e^-$ pair production is
estimated based on the JWKB approximation. It is found that most positrons are
emitted at $t\lesssim 10^{-15}$ s, while a long lasting positron emission is
observed for large objects with $R\gtrsim 1000$ fm. The emission and
annihilation of positrons from supercritically charged objects may be partially
responsible for the short $\gamma$-ray burst during the merger of binary
compact stars, the 511 keV continuum emission, as well as the narrow faint
emission lines in X-ray spectra from galaxies and galaxy clusters.
