Infrared and nebular lines provide some of our best probes of the physics regulating the properties of the interstellar medium (ISM) at high-redshift. However, interpreting the physical conditions of high-redshift galaxies directly from emission lines remains complicated due to inhomogeneities in temperature, density, metallicity, ionisation parameter, and spectral hardness. We present a new suite of cosmological, radiation-hydrodynamics simulations, each centred on a massive Lyman-break galaxy that resolves such properties in an inhomogeneous ISM. Many of the simulated systems exhibit transient but well defined gaseous disks that appear as velocity gradients in [CII]~158.6$\mu$m emission. Spatial and spectral offsets between [CII]~158.6$\mu$m and [OIII]~88.33$\mu$m are common, but not ubiquitous, as each line probes a different phase of the ISM. These systems fall on the local [CII]-SFR relation, consistent with newer observations that question previously observed [CII]~158.6$\mu$m deficits. Our galaxies are consistent with the nebular line properties of observed $z\sim 2-3$ galaxies and reproduce offsets on the BPT and mass-excitation diagrams compared to local galaxies due to higher star formation rate (SFR), excitation, and specific-SFR, as well as harder spectra from young, metal-poor binaries. We predict that local calibrations between H$\alpha$ and [OII]~3727$\backslash AA$ luminosity and galaxy SFR apply up to $z>10$, as do the local relations between certain strong line diagnostics (R23 and [OIII]~5007$\backslash AA$/H$\beta$) and galaxy metallicity. Our new simulations are well suited to interpret the observations of line emission from current (ALMA and HST) and upcoming facilities (JWST and ngVLA).