The three-component ionic microemulsion system consisting of AOT/water/decane shows an unusual phase behaviour in the vicinity of room temperature. The phase diagram in the temperature-volume-fraction (of the dispersed phase) plant: exhibits a lower consolute critical point at about 40 degrees C and 10% volume fraction. A percolation line, starting from the vicinity of the critical point, cuts across the plane, extending to the high-volume-fraction side at progressively lower temperatures. This phase behaviour can be understood in terms of a system of polydispersed spherical water droplets, each coated by a monolayer of AOT, dispersed in a continuum of oil. These droplets interact with each other via a hard-core plus a short-range attractive interaction, the strength of which increases with temperature. We show that Baxter's sticky-sphere model can account for the phase behaviour, including the percolation line, quantitatively provided that the stickiness parameter is a suitable function of temperature. We use the structure factors measured by small-angle neutron scattering (SANS) below the critical temperature to determine this functional dependence. We also investigate the dynamics of droplets, below and approaching the critical and percolation points, by dynamic light scattering. The first cumulant and time evolution of the droplet density correlation function can be quantitatively calculated by assuming the existence of polydispersed fractal clusters formed by the microemulsion droplets due to attraction. The relaxation phenomena observed in an extensive set of measurements of electrical conductivity and permittivity close to percolation can also be interpreted through the same cluster-forming mechanism, which reproduces the most relevant features of the frequency-dependent complex dielectric constant of this system.