Rabbits don't cry. They can grieve the loss of a companion, but that would be more lack of activity and appetite. If he is making sounds that seem like he is crying, then it's going to be some sort of health condition causing the vocalization you are hearing. It could be a respiratory problem or restricted air passages, it could be pain causing him to make sounds, or he could have a heart condition or a mass. Then with him having a protruding third eyelid, that can also be an indication of a heart condition or a mass in the chest(thymoma). I suspect he has something more serious than a uti or stress causing all of these symptoms you are seeing, which could account for the urinating and sitting in the puddle, the decreased appetite, the bulging third eyelid, and the noisy breathing and increased respiratory effort. All of those symptoms would make me suspect a heart problem or thymoma, unless chest xrays were done that ruled out these possibilities. It's a possibility I mentioned at the beginning of this thread, and with the additional symptoms presenting, it seems even more possible now.
For a rabbit with serious preexisting medical issues, it is going to be difficult to rehome him. You won't be able to travel any distance without the stress of the journey posing significant risk. So you would have to find someone local willing to take him on and his potential medical issues, but then you have to consider the stress it puts on him relocating to an unfamiliar place.
To figure out what's going on with him is likely going to involve a lot of expensive tests. And even if the vet can determine a cause, it may not be treatable. Then there is considering if the costs of further testing is even doable for you. Honestly, it sounds to me like he is not doing well and is likely suffering from significant pain of some sort. As hard as it is, it might be time to consider what is best for him at this point and his quality of life. He's lost his companion and he's unwell with a significant undiagnosed illness. Sometimes even if treatment is possible, it's not always the best option if it just prolongs the suffering of the animal. Quality of life is a very important consideration. Think of what his quality of life is now and what it will be as time progresses. Is he happy, does he still enjoy doing things, is his pain managed/managable so he can be comfortable for his remaining time? These are all important things to evaluate to determine what the next step should be for you and your bun.
