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Most of the "lore" comes from talking to merchants, and the descriptions you get when mousing over the ship type in the hangar. The story, especially concerning the vault of souls and the commander, is really open ended and vague, in my opinion. 

The developers are independent and not associated with Subset. They don't hide the FTL influence, and have posted several times in r/ftlgame advertising Void War (among other subreddits). If I had to speculate on why they don't make references to FTL in the marketing material and FAQs, it's probably because they want their game to stand on it's own, rather than being a derivative game "for fans of FTL."

One of the developers in the discord said "the marketing for this game has been kind of a nightmare [...] I kinda wonder if people just need some time to get used to the idea that something so directly inspired by ftl could actually be good. I'm pretty confident that quality will win in the end. Just gotta give it some time"

There are significant gameplay differences. TL;DR void war has way more crew, an equipment system, a larger focus on crew combat,  more ship armament and system types, and way more difficulty options. 

While FTL has 8 crew types, Void War has 99 (at the time of writing), which each belong to 7 or 8 different factions with unique characteristics (all Death Cultists are poison immune, almost all Blood Cultists interact with thralls in some way, and so on). Most crew also have different amounts of equipment slots where they can equip different armor, weapons, psychic abilities, and so on. 

The game is more focused on crew combat: units with the ability to teleport around are common, boarders are plentiful, and since each run starts with you choosing a powerful Commander unit, you can start boarding right away.

Or you can summon Ghouls on the enemy ship to board for you. Or you can send your Sub-Husks straight from the factory into the assault sled to overwhelm the enemy with numbers. Or all three at once.

Aside from that, there are a ton of different systems, Missile and Siege/Artillery weapons work differently, traversing through the sector works more like STS than FTL, the Lance weapon type is around, there's 6 sectors instead of 8, there's 3 bosses instead of 1, and there are 13 difficulty options instead of 3.

Spoilers below:
-For your fourth bullet point, you can do both. I believe you just have to loot his hud from an OR on a later day.
-Defeating the bartender doesn't affect the veterinarian's quest. You can get her as a party member if you snoop around in the detective's office, find incriminating evidence, blackmail her, and then defeat her in combat.

As a heads-up, after the end of the Engineering vs Atmosia quest, if you attempt to sleep, you will get softlocked on an "end of current content" screen where you're not able to save. Save before then if you want to finish the sidequests.
Also, right clicking on the "Victory" screen after combat is finished results in a softlock. Also, trying to complete the lawyer's quest after some time has passed has strange results, like bodies disappearing and certain encounters with the securitrons or the clown being nonexistent.