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We perform a stringent search for precursor emission of short gamma-ray bursts (SGRBs) from the Fermi/GBM data and find 16 precursor events with $\gtrsim4.5σ$ significance. We find that the durations of the main SGRB emission ($T_{\rm GRB}$) and the precursor emission ($T_{\rm pre}$), as well as the waiting time ($T_{\rm wt}$) in between, are roughly comparable to each other, with $T_{\rm wt}\approx2.8T_{\rm GRB}^{1.2}$ approximately satisfied for most cases except one significant outlier. We also perform spectral analyses to the precursors and SGRBs, and find that the spectra of precursor emission can be fitted with the blackbody, non-thermal cutoff power law and/or power law models. We consider several possible models for precursor emission in SGRBs and find that the luminosity and spectral shape may be explained by the the shock breakout or the photospheric radiation of a fireball launched after the merger for thermal precursors, or magnetospheric interaction between two NSs prior to the merger for non-thermal precursors. For the fireball photospheric model, a matter-dominated jet is preferred and a constraint on the fireball Lorentz factor can be placed as $Γ\sim30$. For the magnetospheric interaction model, jet launching mechanism may be constrained. In particular, those events with $T_{\rm wt}/T_{\rm GRB}\gg1$ (e.g. GRB191221802) require the formation of a supramassive or stable neutron star after the merger, with the delay time defined by the timescale for an initially baryon-loaded jet to become magnetically dominated and relativistic.