We experimentally demonstrate the two-stage structural and slowing-down percolating transitions, followed by the confluent transition in the densifying cancer cell monolayers from the dilute state, and investigate their impacts on collective cell dynamics. It is found that cells aggregate into clusters at low cell density. With increasing cell number density, the structural percolation through the formation of a large cell cluster percolating through the space precedes the dynamical percolation transition of forming a percolating cluster of slow cell elements. Both percolating transitions exhibit scale-free scaling behaviors of cluster size distributions and fractal structures, similar to those of the universality class of 2D nonequilibrium systems governed by percolation theory. Dynamically, at low cell density, cell aggregation enhances cooperative motion. The structural percolation leads to slower motion, especially with stronger suppression for the high-frequency modes in the turbulent-like velocity power spectra. The following slowing-down percolation associated with the onset of cell crowding in regions occupied by cells further enhances dynamical slowing-down, and suppresses the increasing trend of dynamical heterogeneity and the steepening of the power spectrum of motion, until their reversions after the confluent transition.