I'll see if I can clear up some stuff, considering I already nerded out about theology in Griftlands.
--- WARNING! WARNING! HUGE LEVELS OF NERD INCOMING! ---
a)This is a pretty old question, I remember asking about this last year. As far as we know the Feud has not been classified in any logs so far. I can at least clear up one thing, it's probably NOT the name of the planet. Quotes like 'Another day in the Feud, hm?' imply through semantics that the Feud is either an organisation, class of people, or a specific location upon the planet, not the planet itself.
Here's most of the references to the Feud I can find in game:
The only real lead we have on this is a random labourer battle cry 'The Feud will rise!', it's specifically FOR the labourer NPC class. I put my money on the Feud being an informal term for areas where the lower classes of the world live, kinda like slums???
I can only assume that the relationship quote is a very old piece of writing, it wasn't made with the knowledge it'd be used for every single NPC.
b) Sal's campaign takes place in Murder Bay. That we know for sure, New Deltree is the foothold base of operations for the colonising Admiralty into Havaria. Judging by its name and the abundance of the Spree, Murder Bay was probably a loosely organised union of small settlements before the admiralty set up, using their massive military budget to construct what the game calls 'The optimistically-named "capital" of Admiralty-occupied Havaria, and the biggest city in Murder Bay.- the way these qoutes are worded implies that the area Admiralty has control over in Havaria is very small. Sal's campaign will take her to New Deltree, to do quests and such, but the bulk of it is spent in small settlements around Murder Bay, like the Grog'n'Dog.
c) Firstly, we can confrim that Supplicant City and Rentoria are near each other, mostly because Heshies and Rentorians hate eachother and have most likely fought at some point.
We can also assume that Supplicant city is HUGE, the Cult is shown to be very old, possibly going back to the Vagrant Age. So they would have the time (and resources considering they basically own all Lumin and the Ik-Derricks that work it) to build a huge city. On the other hand, judging by Pearl's class of people, it was probably founded to be a relaxing getaway for the 'cutthroat' Heshies in Supplicant. I can only imagine that Supplicant city is a vastly more strict version of Pearl-on-the-Foam, considering Moreef mentions that devote Heshians don't drink alcohol, and the most devote Heshians come from Supplicant City.
As for the reasons why Havaria is being settled. Firstly, the Cult definitely wasn't the first group to stumble upon the contient considering Vagrant Age tech exists in abundance there. It's pretty simple as to why they'd first colonise it though;
Havaria is a continent,
Continents have a lot of coastline,
Creatures blessed by Hesh wash up on coastlines constantly.
The Admiralty most likely got to Havaria a lot later, as they had to found Deltree itself first. The Admiralty again have a pretty clear motive for colonising Havaria, they're a military nation that needs subjects to rule over, and they need fuel and resources to fund themselves. There was aparrently a Rento-Admiralty war at some point in time so I can imagine the recent creation of New Deltree is to help the Admiralty recoup losses from that war.
As for WHY the Admiralty hasn't stepped on Grout Bog and Supplicant City, like I said they've only recently made a stand in Havaria with New Deltree. Hesh, even Sal's story is about the Admiralty trying to quash the last of the Spree in Murder Bay so they can keep the area for themselves. This would also explain why the Admiralty aren't a big fear for most of Havaria - they just don't currently have the respect or age to make Havarians obey them.
For the Bog, it has been consitently described as a impassible and deadly swamp that drives most people away. The Admiralty probably doesn't want to bother spending a ton of resources just trekking through the damn thing to get to talk to the owners of it, the Spark Barons. Not to mention the Boggers basically cornering off a large chunk of the place.
Speaking of the Barons, they're not a nation or a religious organisation really - they're just a corporation with an absolute ton of Shills to spend. This is already a problem for the clearly weak-to-bribes Admiralty. Not only do they dig up the Grafts used by people, but they're also the sole producers of spark. The Admiralty has good reason not to anger the people providing them with all of their Grafts, and a likely bulk of their fuel (as spark has been described as a much cheaper and popular alternative to lumin).
The Bog is basically one big company store, where the barons themselves seem to enforce law over their workers - they're a closed economy basically.
Smith's story does show that the Rise and Admiralty are in a feud with eachother too, so it's again a good decision for the Admiralty military to not risk stepping into the civil war between the Rise and the Barons when both of them have reasons to turn on the Admiralty.
As for Supplicant City, it's pretty clear that Hesh is the main religion in Havaria. If the Admiralty were to invade Supplicant City, they would be attacking the religion of their citizens (not to mention the clear military might the cult has). The quote 'The symbol of the Deltrean Empire and its lineage. When the Cult of Hesh became the dominant religion, the "many roots" of the Deltrean Tree became synonymous with Hesh's many tentacles. ' clearly implies that the Admiralty themselves are mostly worshipers of Hesh too, they probably wouldn't want to damage their religion's holy land. They are also the sole owners of Lumin, so going against them would mean losing a ton of your fuel sources. Sal's story does show that they have been trying to achieve some soft power over the Cult with the auction at the end of the day 5, though what became of that we don't know.
The Admiralty's existance in Pearl-on-the-Foam is likely due to its previously mentioned tourism industry (there's reasons to visit it, unlike the Bog). There's clearly a lot of crime going on (as there's more than enough rich suckers to make a living off of), but the Cult aren't lawmakers - they only go after people who go against Hesh's doctrine. They probably don't mind the Admiralty setting up a police office in their smaller vacation city and keeping their pretty little holiday location clean, the Cult still clearly holds rule over them there with their overwhelming dogma.
The Breadfields are rightfully a top priorty for the Admiralty - an army marches on its stomach after all.
Finally, the Roaloach doesn't seem to be a place that the Bile Brokers actually live at, rather its a dangerous enivroment that only they can work in because of their equipment and training. The Admiralty can just buy the resources they need from the Bile Brokers, as they aren't a large group of people. Its just a small archipelago after all.