We report measurements of the branching fractions and direct CP asymmetries (A(CP)) for B -> K pi, pi pi and KK decays (but not pi(0) pi(0)) based on the final data sample of 772 x 10(6) B (B) over bar pairs collected at the Y(4S) resonance with the Belle detector at the KEKB asymmetric-energy e(+)e(-) collider. We set a 90% confidence-level upper limit for K+K- at 2.0 x 10(-7); all other decays are observed with branching fractions ranging from 10(-6) to 10(-5). In the B-0/(B) over bar (0) -> K-+/-pi(-/+) mode, we confirm Belle's previously reported large A(CP) with a value of -0.069 +/- 0.014 +/- 0.007 and a significance of 4.4 sigma. For all other flavor-specific modes, we find A(CP) values consistent with zero, including A(CP)(K+pi(0)) = +0.043 +/- 0.024 +/- 0.007 with 1.8 sigma significance. The difference of CP asymmetry between B-+/- -> K-+/- pi(0) and B-0/(B) over bar (0) K-+/-pi(-/+) is found to be Delta A(K pi) equivalent to A(CP)(K+pi(0)) - A(CP)(K+pi(-)) = +0.112 +/- 0.027 +/- 0.007 with 4.0 sigma significance. We also calculate the ratios of partial widths for the B -> K pi decays. Using our results, we test the validity of the sum rule A(CP)(K+pi(-)) + A(CP)(K-0 pi(+)) Gamma(K-0 pi(+))/Gamma(K+pi(-)) - A(CP)(K+pi(0))2 Gamma(K+pi(0))/Gamma(K+pi(-)) - A(CP)(K-0 pi(0))2 Gamma(K-0 pi(0))/Gamma(K+pi(-)) =0 and obtain a sum of -0.270 +/- 0.132 +/- 0.060 with 1:9 sigma significance. DOI: 10.1103/PhysRevD.87.031103