We report a Dalitz plot analysis of charmless hadronic decays of charged B mesons to the final state KS0π+π0 using the full BABAR data set of 470.9±2.8 million BB events collected at the ϒ(4S) resonance. We measure the overall branching fraction and CP asymmetry to be B(B+→K0π+π0)=(31.8±1.8±2.1-0.0+6.0)×10-6 and ACP(B+→K0π+π0)=0.07±0.05±0.03-0.03+0.02, where the uncertainties are statistical, systematic, and due to the signal model, respectively. This is the first measurement of the branching fraction for B+→K0π+π0. We find first evidence of a CP asymmetry in B+→K∗(892)+π0 decays: ACP(B+→K∗(892)+π0)=-0.52±0.14±0.04-0.02+0.04. The significance of this asymmetry, including systematic and model uncertainties, is 3.4 standard deviations. We also measure the branching fractions and CP asymmetries for three other intermediate decay modes.