@David
do horses, cats, dogs, pet rats, etc. Exist in the Persean Sector?
I haven't found a need to answer this. Consider it an unknown that you can project into.
Is medicine a big deal in the Persean Sector? What kind of healthcare system does the Hegemony use for example? I assume Tri-Tachyon has privatised healthcare obviously. But what about Persean League, Hegemony or even the Sindrian Diktat?
Regarding details, see above answer.
Nonetheless, I would imagine them as extensions of the core themes & aesthetics of the faction. This is not to say that all doctors in Hegemony space are militarized - each faction draws the line of purview of its state / military-acting-like-a-state at a different place. Beyond that, the civic structures of planets within the Persean League and Hegemony can vary quite a bit between worlds, so I wouldn't say there is a standard answer - unless we're talking specifically about military hospitals, which would be standardized in each, and especially in the Hegemony.
The Diktat is smaller and more extreme than either, but medical administration isn't at any point a key aspect of Andradan ideology (beside some weird and not medically valid talk about idealizing 'strength/purity' and the like), so I would think they would inherit and adapt a combination of civilian structures that existed before the Diktat as well as a modified Hegemony structure on the military side.
"Doctor" Cydonia might have some thoughts on all of this, of course.
Is food retrieved from ancient stations that has infinite shelf life that much worse than fresh produce in taste and nutritional value?
I would not imagine it's better than the fresh stuff, no. Then again, your common spacer likely doesn't have much experience with good, fresh food.
Is fishing popular on Eochu Bres?
I have no idea.
Are there are other kinds of mobile devices than the TriPad? Is there a Domain's equivalent of Linux?
I've mentioned the existence of an "off-brand datapad" here and there, so there are others. All the cool kids have TriPads though.
Is the Domain the only real major human political entity or are there are other hiding in the dark whispered about? Ones that succeeded in defeating the Domain and aren't just occupying unwanted backwaters? (this might be a spoiler, though I guess it won't matter, why would they ever come to the Persean Sector).
All we know is what the Domain said, and they said they were #1 so I guess that's your answer

Were there ever any high-tech XIV modified ships, before the long journey to the Persean Sector?
If it's not in the game, we don't know if it existed or not.