You have described to me a 3 year old male cat who has been sticking his tongue out and has halitosis (bad breath). He is eating normally, not drooling, and is maintaining his weight. He came from a shelter-type background, and was tested for FeLV and FIV before you adopted him. His partner cat had the same symptoms and died suddenly with no diagnosis made as to why, though he did have symptoms of an upper respiratory tract infection before he died.
The top concern I have for your cat is that he may be positive for FIV or FeLV. Both of these are viruses that are only a problem in cats (not dogs, not humans, so your daughter is at no risk). With these viruses, if we do a blood test within 6 months of the cat having been exposed to the virus, the test will come up negative. So, doing a repeat test 6 months later may be necessary. These viruses are not transmitted by casual contact, so you do not need to worry about passing them to the cats you foster via your clothing.
Another possibility is that your cat may have FIP (feline infectious peritonitis) which is also a virus. It is much harder to test for, and we don't have a simple blood test that identifies it. It is also only a problem in cats (not humans) and not transmitted by casual contact.
Let me explain more about FeLV and FIV and FIP.
1 Feline Leukemia Feline leukemia (FeLV) is a devastating virus for which there is no cure once cats are exposed. There is a vaccine for it, which is highly protective, though not completely foolproof. Transmission occurs through infected saliva, bites by or sharing bowls with infected cats. Symptoms are numerous including fever, frequent infections, weight loss, depression, decreased appetite, and swollen lymph nodes. Prevalence of the disease is worldwide with locally high numbers of incidence possible in infected groups of feral cats.
Blood tests can identify infection. Supportive care is the only option for treating cats positive for feline leukaemia; prevention is the therefore the best solution. Cats should be tested and vaccinated if owners intend to allow them outside. If owners are intending to keep cats indoors with no potential for exposure to cats outside of the household, cats need not be vaccinated against feline leukaemia. However, an initial blood test upon bringing a new kitten or cat into the household is recommended to identify whether the cat is carrying the virus.
2. Feline immunodeficiency virus (FIV) is yet again, a seriously destructive and fatal virus for which there is no cure. FIV exists throughout the USA and Canada and is transmitted through bite wounds. There is no standard vaccine to prevent FIV.
Once the virus infects a cat, the cat may live a relatively normal life for many years.
Since the virus affects the immune system of the cat, the cat is less able to fight off infections of any sort and will require supportive care as needed. Symptoms include fever, recurring infections and illnesses, weakness, depression, and weight loss. Prevention is best achieved through minimizing potential exposure to potential carriers.
Never allow direct contact with other cats and practice good hygiene and disinfecting practices. There is currently no known correlation between FIV and human immunodeficiency virus (HIV).
3. Feline infectious peritonitis (FIP) is another serious and destructive virus. The disease is most commonly seen in facilities housing large numbers cats, such as catteries and animal shelters. Transmission occurs when a cat comes into contact with an infected cat's bodily secretions, primarily saliva and feces. Unfortunately, the virus can survive a long time outside of the body and can remain a source of infection on a dirty food bowl or litter pan.
Initial symptoms include upper respiratory problems, depression, and weight loss. Two types of the disease are recognized. "Wet" type FIP-infected cats appear with large "pot-bellied" abdomens that are actually filled with fluid, eventually leaving the cat struggling to breathe. "Dry" type FIP-infected cats have minimal fluid accumulation and exhibit weight loss, depression, anaemia, and fever.
Unfortunately, FIP is hard to diagnose as test results are unreliable; by the time symptoms are identified as likely resulting from FIP, the disease has already significantly progressed. The only way to care for an FIP-positive cat is to provide supportive care based upon the symptoms. A vaccine does exist for this virus but is quite controversial and is not frequently used. There is some concern that not only does it not prevent the disease, but it may actually cause it. The best prevention is to minimize a cat's possibilities of exposure.
From what you are describing about the previous kitty who died suddenly, I wonder whether he may have had one of these viruses. Then, if he was exposed to a simple upper respiratory tract infection (a virus similar to a cold in us), he may not have had a strong enough immune system to fight it.
With your cat who is now sticking out his tongue and having bad breath, he may have an infection in his mouth causing this behaviour and odour. A normal cat might be able to fight this off easily, but if your boy has one of the above viruses, his immune system may not be able to fight it. A course of antibiotics might be needed to help him get over this! Also, doing another blood test to determine if he is now positive for FeLV or FIV would be prudent. A physical exam and blood test called a corona titre would be helpful in deciding if he might have FIP, though the tests are not as definitive as the FeLV and FIV tests.
So, in summary, I am concerned about an underlying virus leading to an inability to fight a bacterial infection. Antibiotics might be all your boy needs! I would strongly recommend you see a veterinarian to determine if this is what is needed to help him get over this.