Regarding organ donations-- one thing you need to know is that most of the major organs (heart, kidney, liver) have to be harvested while the person's body is still technically "alive." In other words, they have to be pronounced brain dead while in a hospital setting and be on a ventilator etc. Heart valves, corneas, and long bones have to be harvested within 24 hours of when the person was last known to be alive. (And this is why it is so important to be an organ donor. The circumstances that make it possible to donate major organs are so rare that actual useful donations become even rarer.)
Regarding pauper burials: In Louisiana, it is generally up to each Coroner's office to decide when/if a body receives a pauper burial. In our parish there is one local funeral home that provides a graveyard for pauper burials. Unclaimed bodies are never cremated. The office here has held bodies for over four months waiting to hear from family regarding funeral arrangements (and it's even more complicated when it's someone from outside the country.) If a body is turned over for pauper burial, the grave is recorded with GPS coordinates, but there is no headstone and the location is not released to the family. This is to avoid having cheapo families allowing the state to pay for the funeral. Of course families can still get the location through Vital Statistics, but most people don't realize that.
Also, something to think about in reference to bodies being held until families can be found: bodies in a morgue cooler are still decomposing, just not as quickly as out in the open. Morgue coolers are NOT freezers. Even after just a few days in a cooler, a fresh body will have marbling and the beginnings of decomp. After four months in a cooler, that body is liquid mush.