Let $M$ be a conected, nonorientable 3-manifold, with a homology class $[K] \in H_{2}(M; \mathbb{Z} / 2 \mathbb{Z})$ and a class $[\gamma] \in \pi_{1}(M)$ which have zero mod-2 algebraic intersection: $[K] \cdot [\gamma] = 0 \in \mathbb{Z} / 2 \mathbb{Z}$.
1) Can one find representatives for these classes which have zero geometric intersection?
2) If so, can one choose them such that the representative for $K$ is the mod-2 fundamental class of an embedded surface?
If $[K]$ is represented by some embedded surface $K$ and $[\gamma]$ by a loop $\gamma$ that intersects $K$ transversely in an even number of points, then consider two adjacent points of intersection on $\gamma$. You can use the stretch of $\gamma$ between them to do surgery on the surface, cutting two small holes around the intersection points and then attaching a tube along the stretch of $\gamma$ that connects them. This surface is homologous to the original $K$, and intersects $\gamma$ in two fewer points. Repeat until there are 0 points of intersection.