We measure the branching fractions and CP asymmetries in the decays B0-->K(+)K(-)K(0)(S) and B+-->K(+)K(0)(S)K(0)(S) using a sample of approximately 122x10(6) BB pairs collected by the BABAR detector. From a time-dependent analysis of the K(+)K(-)K(0)(S) sample that excludes phiK(0)(S), the values of the CP-violation parameters are S=-0.56+/-0.25+/-0.04 and C=-0.10+/-0.19+/-0.10, where the first uncertainty is statistical and the second is systematic. We confirm that the final state is nearly purely CP even and extract the standard model parameter sin(2beta=0.57+/-0.26+/-0.04(+0.17)(-0) where the last error is due to uncertainty on the CP content. We present the first measurement of the CP-violating charge asymmetry A(CP)(B+-->K(+)K(0)(S)K(0)(S))=-0.04+/-0.11+/-0.02. The branching fractions are B(B0-->K+K-K0)=(23.8+/-2.0+/-1.6)x10(-6) and B(B+-->K(+)K(0)(S)K(0)(S))=(10.7+/-1.2+/-1.0)x10(-6).