Charmed hadron production has been studied in 225 pb^-1 of data collected with the ZEUS detector in the HERA-II runnning period. Studies were made in the D* → D⁰π → Kππ decay channel to determine the potential for the microvertex detector to improve charm signals. Integrated cross sections were measured for the decays D⁰ → K⁰_sπ⁺ π⁻, D^± → K⁰_s π^±, D^±_s → K⁰s K^± and Lambda^±_c → Lambda⁰π^± in the kinematic region 5< Q² < 1000 GeV², 0.02 < y < 0.7, 1.5 < pT(D,Lambda_c) < 20.0 GeV, -1.6 < pseudorapidity(D, Lambda_c) < 1.6. The D⁰ cross section was additionally measured down to pT(D⁰) > 1.0 GeV. Differential cross sections as a function of Q², x, pT(D) and pseudorapidity(D) were measured for the D⁰, D^±, and D_s^± channels. The measurements were compared to next-to-leading order (NLO) QCD calculations, and good agreement found for all variables in all channels over the measured kinematic region. The strangeness-suppression factor, gamma_s = D_s^±/(D^± + D⁰), was measured and found to be in good agreement with values previously measured at HERA and elsewhere, and with the DJANGOH and RAPGAP Monte Carlo. The ratio of charmed baryons to mesons, Lambda_c^±/(D^± + D⁰), was measured and also found to be in good agreement with Monte Carlo.