We present measurements of branching fractions and CP-violating asymmetries in decays of B mesons to two-body final states containing a K0. The results are based on a data sample of approximately 88 x 10(6) Upsilon(4S)-->BB decays collected with the BABAR detector at the PEP-II asymmetric-energy B Factory at SLAC. We measure B(B+-->K(0)pi(+))=(22.3+/-1.7+/-1.1)x10(-6), B(B0-->K(0)pi(0)=(11.4+/-1.7+/-0.8)x10(-6), B(B+-->K(0)K+)<2.5 x 10(-6), and B(B0-->K(0)K(0)<1.8 x 10(-6), where the first uncertainty is statistical and the second is systematic, and the upper limits are at the 90% confidence level. In addition, the following CP-violating asymmetries have been measured: A(CP)(B+-->K(0)pi(+))=-0.05+/-0.08+/-0.01 and A(CP)(B0-->K(0)pi(0)=0.03+/-0.36+/-0.11.