  In anticipation of the Euclid Wide and Deep Surveys, we present optical
emission-line predictions at intermediate redshifts from 0.4 to 2.5. Our
approach combines a mock light cone from the GAEA semi-analytic model to
self-consistently model nebular emission from HII regions, narrow-line regions
of active galactic nuclei (AGN), and evolved stellar populations. Our analysis
focuses on seven optical emission lines: H$\alpha$, H$\beta$,
[SII]$\lambda\lambda 6717, 6731$, [NII]$\lambda 6584$, [OI]$\lambda 6300$,
[OIII]$\lambda 5007$, and [OII]$\lambda\lambda 3727, 3729$. We find that Euclid
will predominantly observe massive, star-forming, and metal-rich line-emitters.
Interstellar dust, modelled using a Calzetti law with mass-dependent scaling,
may decrease observable percentages by a further 20-30% with respect to our
underlying emission-line populations from GAEA. We predict Euclid to observe
around 30-70% of H$\alpha$-, [NII]-, [SII]-, and [OIII]-emitting galaxies at
redshift below 1 and under 10% at higher redshift. Observability of H$\beta$-,
[OII]-, and [OI]- emission is limited to below 5%. For the Euclid-observable
sample, we find that BPT diagrams can effectively distinguish between different
galaxy types up to around redshift 1.8, attributed to the bias toward
metal-rich systems. Moreover, we show that the relationships of H$\alpha$ and
[OIII]+H$\beta$ to the star-formation rate, and the [OIII]-AGN luminosity
relation, exhibit minimal changes with increasing redshift. Based on line
ratios [NII]/H$\alpha$, [NII]/[OII], and [NII]/[SII], we further propose novel
z-invariant tracers for the black hole accretion rate-to-star formation rate
ratio. Lastly, we find that commonly used metallicity estimators display
gradual shifts in normalisations with increasing redshift, while maintaining
the overall shape of local calibrations. This is in tentative agreement with
recent JWST data.
