A sample of 7253 $K^\pm\to\pi^\pm e^+e^-(\gamma)$ decay candidates with 1.0% background contamination has been collected by the NA48/2 experiment at the CERN SPS, allowing a precise measurement of the decay properties. The branching ratio in the full kinematic range was measured to be ${\rm BR}=(3.11\pm0.12)\times 10^{-7}$, where the uncertainty includes also the model dependence. The shape of the form factor $W(z)$, where $z=(M_{ee}/M_K)^2$, was parameterized according to several models, and, in particular, the slope $\delta$ of the linear form factor $W(z)=W_0(1+\delta z)$ was determined to be $\delta=2.32\pm0.18$. A possible CP violating asymmetry of $K^+$ and $K^-$ decay widths was investigated, and a conservative upper limit of $2.1\times 10^{-2}$ at 90% CL was established. Show Partial Abstract