Ultraviolet (UV; rest-frame $\sim1200-2000$ A) spectra provide a wealth of diagnostics to characterize fundamental galaxy properties, such as their chemical enrichment, the nature of their stellar populations, and their amount of Lyman-continuum (LyC) radiation. In this work, we leverage publicly released JWST data to construct the rest-frame UV-to-optical composite spectrum of a sample of 63 galaxies at $5.6<z<9$, spanning the wavelength range from 1500 to 5200 A. Based on the composite spectrum, we derive an average dust attenuation $E(B-V)_\mathrm{gas}=0.16^{+0.10}_{-0.11}$ from \hb/\hg, electron density $n_e = 570^{+510}_{-290}$ cm$^{-3}$ from the O II doublet ratio, electron temperature $T_e = 17000^{+1500}_{-1500}$ K from the O III $\lambda4363$/ O III $\lambda5007$ ratio, and an ionization parameter $\log(U)=-2.18^{+0.03}_{-0.03}$ from the O III/O II ratio. Using a direct $T_e$ method, we calculate an oxygen abundance $12+\log\mathrm{(O/H)}=7.67\pm0.08$ and the carbon-to-oxygen (C/O) abundance ratio $\log\mathrm{(C/O)}=-0.87^{+0.13}_{-0.10}$. This C/O ratio is smaller than compared to $z=0$ and $z=2$ - 4 star-forming galaxies, albeit with moderate significance. This indicates the reionization-era galaxies might be undergoing a rapid build-up of stellar mass with high specific star-formation rates. A UV diagnostic based on the ratios of C III $\lambda\lambda1907,1909$/He II $\lambda1640$ versus O III $\lambda1666$/He II $\lambda1640$ suggests that the star formation is the dominant source of ionization, similar to the local extreme dwarf galaxies and $z\sim2$ - 4 He II-detected galaxies. The O III/O II and C IV/C III ratios of the composite spectrum are marginally larger than the criteria used to select galaxies as LyC leakers, suggesting that some of the galaxies in our sample are strong contributors to the reionizing radiation.