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We report a study of the semileptonic decays D+→KS0π0ℓ+νℓ (ℓ=e, μ) based on 20.3 fb-1 of e+e- collision data collected at the center-of-mass energy of 3.773 GeV with the BESIII detector. The D+→KS0π0μ+νμ decay is observed for the first time, with a branching fraction of (0.896±0.017stat±0.008syst)%, and the branching fraction of D+→KS0π0e+νe is determined with the improved precision as (0.943±0.012stat±0.010syst)%. From the analysis of the dynamics, we observe that the dominant K̄∗(892)0 component is accompanied by an S-wave contribution, which accounts for (7.10±0.68stat±0.41syst)% of the total decay rate of the μ+ channel and (6.39±0.17stat±0.14syst)% of the e+ channel. Assuming a single-pole dominance parametrization, the hadronic form factor ratios are extracted to be rV=V(0)/A1(0)=1.42±0.03stat±0.02syst and r2=A2(0)/A1(0)=0.75±0.03stat±0.01syst. Based on the first comprehensive angular and the decay-rate CP asymmetry analysis, the full set of averaged angular and CP asymmetry observables are measured as a function of the momentum-transfer squared; they are consistent with expectations from the standard model. No evidence for violation of μ-e lepton-flavor universality is observed in either the full range or the five chosen bins of momentum-transfer squared.