The Hunter den thing reminds me of Tommy's Matchbook, just fyi!
I've always imagined Hunters having hideaways tucked around the City and yes, they absolutely squabble over who they belong to. For example, RZ has a hidden cubby in the Tower rafters and the only reason it's undisputed is bc he literally throws people out (Tower security has stopped trying to moderate this).
Hunters definitely have walking songs and sea shanties and little ditties that help make travel easier! I feel like many of them listen to music during gunfights because it helps them focus [rather than distract them] and the music only shuts off if things get really, really bad. There isn't a lot of restrictions about what songs can be played where, but whistling or humming at night is inviting trouble.
Hunters also have hidden signals / languages generally shared between fireteam members. "Hunter Sign Language" [HSL] is more widely known since the Lost Sector symbol is part of that language system.
Cloaks are also a big part of Hunter superstition and cloak patterns (canonically personalized to each individual) are used to tell feats or stories. They are often passed down, lent or exchanged as vows, and lost in various gambles. Repairing someone else's cloak is seen as an act of love / closeness and messing with someone's cloak is pretty much akin to fighting words.
I have so many thoughts on Hunter cloaks btw.
In addition to cloaks, Hunters also have trinkets they carry for good luck. Trinkets don't have to be trophies from missions or Crucible; they can be things like mint tins, bullet casings, charms, worry stones, or their favorite pen. Whatever it is, having it is the important part.
Even if a Hunter isn't good at first aid or fixing or macgyvering things, they always have things on-hand that might be useful. Found a full loot cache? Thank your Hunters, bc that's their way of looking out for each other in the wilds. Have some scrap metal or spare parts that no longer fit in your pack? Leave them in loot caches for the next Hunter.
I like to think there are specific superstitions regarding subclasses as well. Like, if you have a Nightstalker on your team, you have to let them scout ahead or you'll encounter an Ogre (even if there aren't any Hive in the area). A Gunslinger's first goldie determines how well [or badly] your assignment will go. If an Arcstrider's grenade or super sparks a certain way, that's definitely an omen for the weather.
I have more headcanons about Hunter culture, but yeah! Hunters!! :D