I have some questions first for you.
How old is your dog?
Does he get into anything around the house? trash?
When he is outside is he always supervised? Can he get out of your yard?
Is he on any medications?
Has he ever had any other health problems?
What type of thyriod condition? Dogs are usually low or hypothyroid?
Do you know what his level is?
Why is he not being treated for that?
Do you think he is having any trouble seeing?
What do his pupils look like?
Are they even?
Has he had any urinary accidents/fecal accidents recently?
Done anything else out of character?
In regard to a dog walking backwards I would want to know if they are having trouble seeing (which you checked pretty well.)
Are they in pain? You said you use rimadyl and usually painful animals are not confused.
The last big area of interest would be neurologic. Confusion would also fit as well as walking backwards. Untreated/severe hypothyroidism in dogs can lead to seizures and neurologic problems. Hypothyroid dogs have been described as having a "tragic expression" the sad face you described. This can be due to poor muscle tone to the face and poor tone is usually related to a neurologic problem.
I would recommend calling the veterinarian and suggesting a recheck of his thyroid level as well as a thorough exam. Because this is not the only diagnosis that fits what you are seeing. But from the history you provided it seem likely. Don't wait too long as I would worry about him possibly having a seizure.
Untreated hypothyroidism can also cause wt gain and poor immune function. Maybe you can get more information that will convince all parties involved to get him on treatment.
If I can answer anything else please let me know.