We report observations of the b --> d penguin-dominated decays B+ --> K0K+ and B0 --> K0K0 in 316 fb(-1) of e+ e- collision data collected with the BABAR detector. We measure the branching fractions B(B+ --> K0K+) = (1.61+/-0.44+/-0.09) x 10(-6) and B(B0 --> K0K0 = (1.08+/-0.28+/-0.11) x 10(-6) and the CP-violating charge asymmetry A(CP)(K0K+) = 0.10+/-0.26+/-0.03. Using a vertexing technique previously employed in several analyses of all-neutral final states containing kaons, we report the first measurement of time-dependent CP-violating asymmetries in B0 --> K(S)0K(S)0, obtaining S = -1.28(-0.73-0.16)(+0.80+0.11) and C = -0.40+/-0.41+/-0.06. We also report improved measurements of the branching fraction B(B+ --> K0 pi+) = (23.9+/-1.1+/-1.0) x 10(-6) and CP-violating charge asymmetry A(CP)(K0 pi+) = -0.029+/-0.039+/-0.010.