  We report a study of $\Lambda_c^+\to\Lambda h^+$ and
$\Lambda_c^+\to\Sigma^{0} h^+$ ($h\!=\!K,\,\pi$) decays based on a data sample
of 980~${\rm fb}^{-1}$ collected with the Belle detector at the KEKB
energy-asymmetric $e^+e^-$ collider. The first results of direct $C\!P$
asymmetry in two-body singly Cabibbo-suppressed (SCS) decays of charmed baryons
are measured, $A_{C\!P}^{\rm{dir}}(\Lambda_c^+\to\Lambda
K^+)\!=\!+0.021\pm0.026\pm0.001$ and
$A_{C\!P}^{\rm{dir}}(\Lambda_c^+\to\Sigma^0K^+)\!=\!+0.025\pm0.054\pm0.004$. We
also make the most precise measurement of the decay asymmetry parameters
($\alpha$) for the four modes of interest and search for $C\!P$ violation via
the $\alpha$-induced $C\!P$ asymmetry ($A_{C\!P}^{\alpha}$). We measure
$A_{C\!P}^{\alpha}(\Lambda_c^+\to\Lambda K^+)\!=\!{-0.023\pm0.086\pm0.071}$ and
$A_{C\!P}^{\alpha}(\Lambda_c^+\to\Sigma^0K^+)\!=\!{+0.08\pm 0.35\pm 0.14}$,
which are the first $A_{C\!P}^{\alpha}$ results for SCS decays of charmed
baryons. We search for $\Lambda$-hyperon $C\!P$ violation in
$\Lambda_c^+\to(\Lambda,\,\Sigma^0)\pi^+$ and find
$A_{C\!P}^{\alpha}(\Lambda\to p\pi^{-})\!=\!{+0.013\pm0.007\pm0.011}$. This is
the first time that hyperon $C\!P$ violation has been measured via
Cabibbo-favored charm decays. No evidence of baryon $C\!P$ violation is found.
We also obtain the most precise branching fractions for two SCS $\Lambda_c^+$
decays, $\mathcal{B}(\Lambda_c^+\to\Lambda
K^+)\!=\!(6.57\pm0.17\pm0.11\pm0.35)\times10^{-4}$ and
$\mathcal{B}(\Lambda_c^+\to\Sigma^0K^+)\!=\!(3.58\pm0.19\pm0.06\pm0.19)\times10^{-4}$.
The first uncertainties are statistical and the second systematic, while the
third uncertainties come from the uncertainties on the world average branching
fractions of $\Lambda_c^+\to(\Lambda,\,\Sigma^0)\pi^+$.
