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 $Agtrsim 315$, $5times10^4$, and $1.2times10^8$, 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 $2.7times10^5$. The $e^+e^-$ pair creation will inevitably start for superheavy nuclei with charge numbers $Zgeq177$, $ud$QM nuggets with $Zgeq163$, strangelets with $Zgeq 192$, and strangeon nuggets with $Zgeq 212$. A universal relation $Q/R_e = left(m_e - bar{mu}_eright)/alpha$ 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 $tlesssim 10^{-15}$ s, while a long lasting positron emission is observed for large objects with $Rgtrsim 1000$ 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.
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