Yes, you need to get your dog to a vet as soon as possible. There are a lot of things that could be going on with the cough, and in a 12-year-old dog it could be something quite serious. Congestive heart failure comes to mind as a possibility, but even a respiratory infection can be a big problem in a senior dog.
Do you think your vet would give you something ahead of time to calm your dog down so he's not so upset about being transported? He really does need to be seen, and soon.
So , here are my thoughts. What you are describing in your dog's right eye is called "Horner's syndrome". It is not a disease in itself, but it does indicate some disruption of the sympathetic nerve/trunk which originates in the first half of the spinal cord and then it runs through the chest cavity and innervates the face. (These are all generalizations, but it essentially supplies nerves to these areas). What this means is, anywhere along this long line, something can be compressing the nerve and it is not able to conduct impulses to things like the eye. Here, you see the dropping of the eyelid and prolapsing of the third eyelid as an indicator. I think the cough is related. I think there could be something as simple as a pneumonia (although not simple) up to cancer in the lungs or mediastinum that is causing the cough. Lung cancer usually is spread from somewhere else, so a chest xray is needed, buy my guess is your dog has cancer somewhere else. Any lumps and bumps on his body ? As we all know, there are exceptions to every rule and I could be slightly off or completely wrong, but this sounds pretty classic and I would put my money on it. Your little guess needs to see a vet to find out exactly what it is and what you can do to treat him/make him more comfortable.
One more thought; a growth in the spinal cord or intervertebral disc disease in the 1st half of the spinal cord can also cause Horner's syndrome.
Thank you heaps for both your answers.
All outcomes you've expressed all scare the heck out of me. I'm going to take him to the vet ASAP. Lucky usually coughs the most after he drinks water. In the day he doesn't often get up, but the few times he does, he takes a drink and it can be a long one. This is where the cough originated. He would cough a few times after drinking a reasonable amount of water. Lucky also gets all his shots in the back of his neck, so could that be numbing/paralysing this nerve you're talking about? We move down his back sometimes when his skin starts to harden around his upper back, as directed by the veterinarian, but since he has little fat on him, the bulk of his mass is towards the upper half of his body. As for the lumps, he does have a few here and there. We got these checked on a previous visit and the vet said they were just fatty tissue lumps. We always try our best to take care of Luck, I just hope we can improve the quality of his life for the time he has left. Thank you again for your answers, any other suggestions or detailed explanations would be appreciated.. just so I can better understand what's going on here and what I might hear on Monday. I'm doing a bachelor of science so don't worry about dumbing it down :)