I must confess that I have not read right through this thread.... I got to about page 8 and became frustrated that none of the original posters updated it with their hen's progress and it just seemed like most people came to the thread, posted a report of their hen's symptoms and disappeared again. 7 years later and people are still coming to this thread and asking the same question. Unless it is in the pages I skipped, I will offer the following suggestions for diagnosis.....
The few people who had young chicks experiencing this almost certainly had omphalitis or pasty butt..... these cases are very different to those suffered by adult chickens.
For those with older laying hens there are a number of possibilities....
A reproductive disorder....
This can be internal laying where the egg yolks released daily from the ovary fail to travel into the oviduct and instead, drop into the abdominal cavity, where they build up over weeks and months to become a large mass. Sometimes the fluid leaches out of them and collects in the lower part of the abdomen with the solidified yolks floating above in amongst the intestines and liver. This build up of fluid and yolk mass puts a strain on the digestive system, respiratory system and the heart. Sometimes an infection will develop in it (Egg Yolk Peritonitis) but often it remains benign but the sheer volume and pressure of it causes the chicken to eventually die. Some do physically split open with the pressure of it, perhaps whilst trying to pass faeces and some suffer heart failure. Usually the bird will appear healthy with a red comb and act normally until the mass gets large enough to put a strain on their organs. They will often walk with a wide stance and may be more upright. During the winter, when they stop ovulating, the condition improves a little as the body is able to reabsorb some of the fluid (the yolks themselves cannot be absorbed or expelled) but in spring when they start ovulating again, their condition will deteriorate. Struggling to breath, particularly in hot weather is common as the mass restricts the exchange of air in the birds rear ventral air sacs. If the bird has been laying eggs recently when the mass is discovered, it is most likely not internal laying..... usually once internal laying starts, they are no longer able to lay eggs normally and for it to become obvious, it has usually been going on for weeks if not months.
The next option is salpingitis..... this is an infection of the oviduct, where egg material and pus build up in the oviduct and become a large lumpy mass. Sometimes the bird will pass some "Lash Egg" material which will be rubbery and have concentric layers of solid material looking like cooked egg and sausage meat when cut open before it gets completely impacted with this stuff and unable to pass anything more. The mass in the oviduct gets so large it constricts the gut and prevents the bird passing faeces properly. Sometimes just white urates can get through and the bird starts to appear sick and go off their food.... this will usually take a matter of a few weeks, maybe a few months. The swelling is more likely to be just below the vent rather than hanging lower down between the legs like internal laying. As the oviduct becomes blocked, the bird will sometimes also start laying internally as well.
Fatty Liver Haemorrhagic Syndrome involves the liver breaking down and leaking fluids into the abdominal cavity. It is sometimes associated with an inappropriate diet and poor exercise. The fluid pools in the belly between the legs similar to internal laying. Draining the fluid can give some relief but the damage to the liver will eventually prove fatal. The bird may continue to lay with this ailment. They may drink more than expected, due to the haemorrhaging. The fluid will also put a strain on the heart and respiratory systems.
A tumour can also cause a swelling of the belly.
Sadly the above illnesses are almost always fatal eventually and the treatment options are limited and expensive ..... the reproductive issues can benefit from regular hormonal implants to prevent ovulation.
The way to find out about these illnesses is to do a post mortem examination(necropsy) to improve your own knowledge. They are sadly all too common as this thread indicates. Some like Fatty Liver Haemorrhagic Syndrome can sometimes be prevented with a better diet and exercise for the remaining flock.... larger DP birds are probably more susceptible to this whereas higher production birds like sex links are more liable to the reproductive issues.