I think that all the haibane (save for the children) died of some suicide.

The ones with the Sin just had some extra extra baggage they were carrying.

why? well its obvious that the city is a mix of Purgatory and limbo

limbo is a place where unbaptized children go to when they die before going to heaven

Purgatory is a place of punishment, all the Haibane are being punished and have to work their way out of it

1)they all have to work
2)they can only wear hand me down clothing and live in run down used places
3) Their names and dreams have something to do with the way they committed suicide
4)The kids didnt commit suicide so they are talking about their dreams in their names no big punishment, just being baptized
5)the people are most likely in Limbo as well, they could be Adults who are not Baptized but didnt commit suicide as they arent punished as heavily as the haibane suicide in Catholicism is a sin so the haibanes having to live with their extra difficulties that the townspeople dont have is a big clue.

Haibane Renmei are the dudes who didnt clear the curse and lost their wings

Toga are the same but somehow received the right to leave the city (renmei mentioned this in like one sentence that I had to rewind to understand what he meant)

on to the suicides

Kuu/Kana
Kuu=Luuli or Air
Lana for Sakana

Both probably killed themselves but drowning

Rakka probably jumped off a building (the crow trying to stop her probably represented her friend trying to stop her from jumping)

Reki Threw herself in front of a train

Nemu(sleep), suicided by overdosing on sleeping pills

Hikari(light) most likely suicided by setting herself on fire

Hyouko(ice lake), jumped into an icy lake

Midori(green) most likely poisoned herself by some kind of chemical. . .
on this, at the time the anime was produced there had been a Rash of suicdes by chemical poisonings over that past couple of years, thing was it was very Dangerous because other people were killed because of it as well due to the fumes it was a big thing in Japan for a while so its no surprise that Abe would have used her as kind of a bit of social commentary there.

more on the toga and renmei
those who are bound by sin for too long lose their wings halos and are forced to live as hermits as further punishment and, work to guide new haibane to purification.

After a while they gain the right to leave the City...unfortunately they still have to carry the mark of a failed haibane which is why they are masked and cannot speak nor touch anyone else