We study the structural and dynamical aspects of 13-atom binary rare-gas clusters of Ar and Xe using constant-energy molecular dynamics simulations. The ground state geometry for ArnXe13-n, n=1-12, remains near-icosahedral, with an Ar atom occupying the central position. The thermodynamic properties of these clusters are significantly different from the pure 13-atom Ar or Xe clusters and for Xe-dominated compositions, melting is preceded by a surface-melting stage. Slow oscillations of the short-time-averaged (STA) temperature are observed both for surface-melting and complete melting stage, suggesting dynamical coexistence between different phases. At the complete melting stage, the oscillations in the STA temperature and the species of the central atom are correlated.