What is the actual emotion of Tzeentch, Nurgle and Slannesh? Why is Nurgle a Chaos God instead of a generic warp deity? What the fuck is Tzeentch even doing? These are all unanswerable questions not by design but by incompetence. GW wants you to believe that there's some sort of overarching thought put into the Chaos Gods but the reality is as I said here and as other people have noticed in the past:
In canon the fear that the gods represent is the degeneration of modern society: you have violent thugs, sick people passing their diseases onto everyone else, weird people with impossible to discern motives and then finally people with different sorts of sexual identities. All it's really doing is swapping the fear from the moral panic of 80s punk culture to a more generalised fear of the world around you. In that swap there's more potential to actually create meaningful duality and frame Chaos as not big bad monsters. Most gods are born out of some sort of fear but by anthropomorphising the fear it enables you to bargain with it to instead produce good outcomes. So while Nurgle might represent a fear of the natural world that also means that you can appeal to Nurgle to clear out your rat infestation or to Khorne to bless you in combat or to Tzeentch for your first date to go well.
That's just how I'd do it though, it's not my thread.
In canon?
Nurgle: Despair. Nurgle's core is all about being ground down by the knowledge that you are going to die. The disease part got tacked on by cultural association.
Tzeentch: Hope. Tzeentch is all about trying to better your place in the universe, by whatever means. The sorcery and scheming comes from cultural associations.
Slaanesh: Selfishness/obsession. Slaanesh was born from the excesses of the Eldar empire, so it has a lot of cruft based on their particular flavor of excess.
NURGLE: oldest and most powerful god in the warp, embodies the state of being alive. Tries to maximize biodiversity, which means all sorts of symbiosis and mutations that most people wouldn't be thrilled about.
KHORNE: embodies the fight-or-flight state of mind that you experience in life or death struggles. In the war torn galaxy of the 41st millennium, he's primarily associated with violent conflict.
SLAANESH: embodies the state of being overwhelmed by physical sensation. Extreme pain, extreme pleasure, anything that puts you into a state of wordless and thoughtless physical experience.
TZEENCH: embodies the state of discontent and aspiring for better circumstances. Tzeench doesn't feed on suffering (that's more Slaanesh's deal), but specifically on the hope that things can get better.
In my approach to Khorne, I went with the idea the gods were shaped by a combination of a primal emotion and a civilization's reaction to that emotion, both positive and negative.
Trying to determine which of three elder gods came first is a moot point: they all started as masses of aggregate emotion. Then civilization came along and started ascribing semi-consistent ideas to those emotions.
Nurgle's primal seed is despair. It picks up disease and deformity from early society's association between those things and despair. It also gets things like acceptance, perseverance and kindness from the various ways society comes up to cope with despair. Those touched by Nurgle know that things will/could get worse, and act based on this. Sometimes that means hastening the end to get it over with, sometimes that means offering comfort where they can.
Khorne's primal seed is fight-or-flight. It picks up violence from one of the common reactions to that state. It also picks up things like honor codes, civility and laws from the ways people have come up to channel those emotions. Those touched by Khorne are driven by a desire to
survive. They usually seek to find a stable pattern of existence, and their reactions when that is threatened are exaggerated.
Nurglite perseverance mutating into Khornate will to live is not entirely unknown. Khornates falling into despair and shifting to Nurgle is a similarly understood phenomenon.
Tzeench's primal seed is ambition. It picks up selfishness and scheming from they ways people have found pursue ambition within society. It also has things like idealism and progress. The defining aspect of those touched by Tzeench is that they have a
vision that they are trying to fulfill.
Slaanesh was born of the ennui/loss of purpose experienced by the Eldar empire. It picked up hedonism from the way some Eldar sought immediate satisfaction to assuage that feeling of meaninglessness. It picked up drive and fanaticism from the attempts of some to find purpose.
Tzeenchians falling into Slaaneshi ennui when they actually manage to achieve a goal and can't find another is common enough. Slaaneshi getting swept up in a Tzeenchian's ambition is also a well known occurrence.
EDIT:
It's pretty clear to me that there's some sort of genetic difference between what I'll call a "racial" deity (the Eldar Pantheon, Gork and Mork and so on) and a Chaos God. The Chaos Gods almost seem like they're meant to be human warp deities but the fact that Slannesh is a specifically Eldar one and that the Chaos Gods alone are capable of drawing power/feeding on all the species of the galaxy makes it clear that there's something special and unique about the big four. They're deliberately set up as more pure/fundamental to the other, more minor warp deities we hear about and honestly the idea that there hasn't always been four Chaos Gods and that they haven't always been locked in an eternal struggle is one of the bigger narrative inconsistencies of 40k for me because of just how omnipresent they are and how universal they're portrayed to be.
So actually maybe I should reverse the question - why is Nurgle a generic warp deity instead of a specific one? Same with the other gods.
Again, in 40k canon, the Eldar gods were
made by the Old Ones. The difference between them and the chaos gods is the difference between a bridge and the giant's causeway.