We investigated the magnetism and superconductivity in as-sintered (AS) and oxidation annealed (OA) T*-type La$_{1-x/2}$Eu$_{1-x/2}$Sr$_x$CuO$_4$ (LESCO) with 0.14 $leq x leq$ 0.28 by the first comprehensive muon spin rotation/relaxation ($mu$SR), magnetic susceptibility, and electrical resistivity measurements. In OA superconducting samples, no evidence of magnetic order was observed, whereas AS semiconducting samples exhibited evidence of a disordered magnetic state in the measured temperature range between $sim$4 K and $sim$8 K. Therefore, the ground state in LESCO drastically varies with oxidation annealing and the magnetic phase competitively exists with the superconducting (SC) phase. The magnetic phase in the AS LESCO is quite robust against Sr doping, while the SC phase degrades with increasing $x$. A monotonous decrease of the SC transition temperature from 24.5 K in $x$ = 0.14 to 9.0 K in $x$ = 0.28 suggests the disappearance of the SC phase at $x$ $sim$ 0.34. Furthermore, we clarified the simultaneous development of (quasi) static magnetism and the electrical resistivity at a low temperature in AS samples, suggesting the inducement of magnetism by the suppression of carrier mobility. The variation in magnetism due to annealing is discussed from a viewpoint of structural defects, which was previously reported from neutron diffraction measurements.
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