Grey feather is purdy

Does being slightly more fragile in chaos gear even matter in a game where your damage resistance already matters very little? The only reason to wear armor is for the unique bonuses anyway.

Chaos is most worth it when it doesn't make things one shot you. This means it is worth it when not fighting fire/shock. We all make mistakes sometimes, being able to tank a single hit if not more is a necessity. It is possible to, utilizing "mainstream weapons" such as dementia praecox suggests, take down most things without taking that single hit, as these weapons tend to benefit greatly from charge time reduction, offer great crowd control, and deal large amounts of damage after being buffed due to their split typing. Max fire/shock UVs can also remove the glass and leave the cannon.

Divine/grey feather are viable for many more levels than chaos, but offer little over things like vog and volcanic demo helm/mercurial demo helm that offer better buffs. Divine veil is the exception, due almost entirely to its fiend med bonus and fire/shock resistance as well as the curiosity that comes with it's split typing. In lockdown, grey feather set is once again outclassed by vog, nameless and other armors- as playing defensively will leave you offensively weaker.

Do you use any of them?
Yes, I do. I use Chaos Set with SSB all the time in FSC or random Arcade runs, I just love the Cloak's looking and yeah of course I love the bonuses screw the negative resistance. I sometimes use Divine veil in Fiend Fire/Shock levels, and I used to use Grey Feather Cloak to be immune to Shock (Like in the GitM) but now I got Mercurial Demo Suit so I use it instead for MSI.

Is it ever more worth it to use one over the others?
Yes, definitely. The Chaos has been the best set since the time when Chaos had only low CTR/DMG but it is now the "bestest" after the buff. Just get Chaos if you wish to save CE to make a lot of sets...well, just get Chaos even if you do have a lot of CE, and use the CE on status UV or something on Chaos.
Despite the Chaos set's awesomeness, other two sets are still good too. Especially the Cowls/Veils look just awesome. Free Halo? Millions of crowns saved. Free Glow Eyes? Millions of crowns saved. ...Magic Line is the nicest line for your wallets in my opinion...

Do the status weaknesses alone ever cause you to put down the Chaos set for another with better resistances?
Well, sometimes. One of the situations would be UFSC and RRT. I don't think being not able to do anything but running for 1 minute is fun, is it? I don't think being shocked and fired for so long at the same time is fun either, is it?

Is there any time when you HAVE to have this gear?
Not necessarily. But well, if someone told me that I had to choose only one gear to do the entire game, the gear would be Chaos set. And if you want to look fancy but want to save money then yeah definitely Divine Veil is the way to go I guess?

What do you think of the looks?
I've been telling this, but all those three sets look terrific. Again, free halo, glow eyes and the chaos cloak's design are making this line just awesome.

Do you think there's a good reason we don't see more Divine Veil and Mantle or even the Grey Feather sets in LD? Curse, fire, and shock resistance, and defense against Elemental and Shadow. Why is it not used much?
I reckon there are several reasons why we don't see Divine/Grey Feather sets in LD.
Firstly, they both don't have offensive bonus agains players; I believe those Skolvers are not Fiends if I'm not blind.
Secondly, the reason why Divine Set isn't used is that it has no Normal defence. If you look closely, Lockdown is flooded with normal damage sources; those Sealed Sword lines, Valiances and even Brandishes (Or Leviathan maybe).
I actually see some bombers wearing those though. Probably because they want to be immune to other haze bombs/Polarises/alchemers.

Any time or experience you've had where this gear or its resistances (or weaknesses) may have saved (or taken) your life?
The Chaos set both saves my life and takes my life away sometimes.
When I'm tired, I tend to stand in front of Freeze Lumbers stupidly with Chaos set+SSB. I get strike, I freeze, looong freeze, can't attack and kill the lumber because I can't turn my body, raise my SSB to protect myself from another freeze, but the shield is a paper shield so it breaks, I'm frozen again, looong freeze, another strike of lumber and so on and so on....I'm dead.
When the set saves me...it is not noticeable but I suppose the CTR has been saving me a lot. In FSC, when I'm wearing Chaos set+SSB I can play roughly quickly and finish FSC very fast, but when I'm wearing something else I can't play that roughly nor that quickly. It's funny though, most people would say that you need to be more careful when waring Chaos+SSB because it's meant to be a glass cannon set, you know. Well I'm guessing it's probably because its CTR max and my habits?