Is there an explicit way of classifying (with regard to their compatibiliy with $\Omega_+$ or $\Omega_-,$ see below) the various families of equivariant holomorphic embeddings from $\mathbb{CP}^1$ to the quintic $Q$, such that $f_*[\mathbb{CP}^1]=2 \in H_2(Q,L;\mathbb{Z})?$

I'd like to know a phenomenology of what one can find, e.g., that only two or three kind of maps would have to be considered, for example the ones giving rise to an open surface, or to an unoriented surface with a crosscap.

Take e.g. a deformation of the Fermat quintic $$ Q = \left\{ (x_1:x_2:x_3:x_4:x_5) \in \mathbb{CP}^4 \left| \quad \sum_{i=1}^5 x_i^5 = \psi \prod_{i=1}^5 x_i \right.\right\},$$ the antiholomorphic involution $$\sigma: \mathbb{CP}^4 \ni (x_1:x_2:x_3:x_4:x_5) \mapsto (\overline{x}_2:\overline{x}_1:\overline{x}_4:\overline{x}_3:\overline{x}_5),$$ $L$ the fixed locus of $Q$ under $\sigma,$ and $\Omega_\pm: \mathbb{CP}^1 \to \mathbb{CP}^1 \ (u:v) \mapsto (\overline{v}:\pm\overline{u});$ equivariance means $f\circ\Omega=\sigma\circ f.$

This should be possible in accordance to Clemens conjecture with $d=2,$ which is proved for small $d$ by Katz (see also Katz2): for a general quintic, we expect from Klemm $n^{g=0}_{d=2}=609250;$ it is known that Fermat quintic is not general in the sense of Clemens conjecture, since we have continuous families of maps.

Here $d$ can be thought more or less equivalently either as the homogeneous degree of the map or as the class in $H_2(Q;\mathbb{Z}).$