We present improved measurements of CP-violation parameters in the decays B0→π+π-, B0→K+π-, and B0→π0π0, and of the branching fractions for B0→π0π0 and B0→K0π0. The results are obtained with the full data set collected at the Υ(4S) resonance by the BABAR experiment at the PEP-II asymmetric-energy B factory at the SLAC National Accelerator Laboratory, corresponding to (467±5)×106 BB̄ pairs. We find the CP-violation parameter values and branching fractions: Sπ+π-=- 0.68±0.10±0.03, Cπ+π-=-0.25±0.08±0. 02, AK-π+=-0.107±0.016-0.004+0.006, C π0π0=-0.43±0.26±0.05, B(B0→π0π0)=(1. 83±0.21±0.13)×10-6, B(B0→K0π0)=(10. 1±0.6±0.4)×10-6, where in each case, the first uncertainties are statistical and the second are systematic. We observe CP violation with a significance of 6.7 standard deviations for B0→π+π- and 6.1 standard deviations for B0→K+π-, including systematic uncertainties. Constraints on the unitarity triangle angle α are determined from the isospin relations among the B→ππ rates and asymmetries. Considering only the solution preferred by the Standard Model, we find α to be in the range [71, 109] at the 68% confidence level. © 2013 American Physical Society.