We present the first results on the Dalitz-plot structure and improved measurements of the time-dependent CP-violation parameters of the process B0→KS0KS0KS0 obtained using 468×106 BB̅ decays collected with the BABAR detector at the PEP-II asymmetric-energy B factory at SLAC. The Dalitz-plot structure is probed by a time-integrated amplitude analysis that does not distinguish between B0and B̅ 0 decays. We measure the total inclusive branching fraction B(B0→KS0KS0KS0)=(6.19±0.48±0.15±0.12)×10-6, where the first uncertainty is statistical, the second is systematic, and the third represents the Dalitz-plot signal model dependence. We also observe evidence for the intermediate resonant states f0(980), f0(1710), and f2(2010). Their respective product branching fractions are measured to be (2.70-1.19+1.25±0.36±1.17)×10-6, (0.50-0.24+0.46±0.04±0.10)×10-6, and (0.54-0.20+0.21±0.03±0.52)×10-6. Additionally, we determine the mixing-induced CP-violation parameters to be S=-0.94-0.21+0.24±0.06 and C=-0.17±0.18±0.04, where the first uncertainty is statistical and the second is systematic. These values are in agreement with the standard model expectation. For the first time, we report evidence of CP violation in B0→KS0KS0KS0 decays; CP conservation is excluded at 3.8 standard deviations including systematic uncertainties.