We perform amplitude analyses of the decays $B^0 \to K^+K^-K^0_S$, $B^+ \rightarrow K^+K^-K^+$, and $B^+ \to K^0_S K^0_S K^+$, and measure CP-violating parameters and partial branching fractions. The results are based on a data sample of approximately $470\times 10^6$ $B\bar{B}$ decays, collected with the BABAR detector at the PEP-II asymmetric-energy $B$ factory at the SLAC National Accelerator Laboratory. For $B^+ \to K^+K^-K^+$, we find a direct CP asymmetry in $B^+ \to \phi(1020)K^+$ of $A_{CP}= (12.8\pm 4.4 \pm 1.3)%$, which differs from zero by $2.8 \sigma$. For $B^0 \to K^+K^-K^0_S$, we measure the CP-violating phase $\beta_{\rm eff} (\phi(1020)K^0_S) = (21\pm 6 \pm 2)^\circ$. For $B^+ \to K^0_S K^0_S K^+$, we measure an overall direct CP asymmetry of $A_{CP} = (4 ^{+4}_{-5} \pm 2)%$. We also perform an angular-moment analysis of the three channels, and determine that the $f_X(1500)$ state can be described well by the sum of the resonances $f_0(1500)$, $f_2^{\prime}(1525)$, and $f_0(1710)$. Show Partial Abstract