Hard to generalise without knowing exactly what you want out of this project, but there are some general pointers based on my experience.
Should I be mapping orthologs after DE and comparing those?
This is probably the simplest and most accurate approach, but also the most limiting. You may need to limit analyses to 1-to-1 orthologs as 1-to-many and many-to-many orthologs can make interpretation considerably more tricky.
Another alternative is just map all the reads to a single reference genome rather than both. Makes interpretation easier, but definitely skews data towards whichever samples are evolutionarily closer to the reference genome you are using. Even within the same species, population level differences affect mapping efficiency.
What do people usually do with species-specific genes that don’t have clear orthologs?
If the whole goal of the project is to compare expression patterns across different species, these genes are not the most useful. More of a footnote where you can describe general patterns where it looks like the species respond differently to their environment. Maybe there is more here, but nothing comes to mind at the moment.
And what’s the usual approach for functional analysis like GO or pathways when working across species?
You can annotate GO terms yourself without too much hassle if you have the compute. The alternative is inherit GO terms or other functional annotations from orthologs if one assembly is annotated and the other isn't. Again, this can get messy and is probably more hassle than just annotating yourself.
