The only thing I'm very certain about is that the vet has checked him for arthritis every time he's gone in, since he is a big, old dog, but he hasn't found any signs of it. I don't know what signs he's checked for, or what you do to rule out arthritis, but the vet doesn't find that to be a concern at this point.
If you really think thyroid function is important to get checked at this point, I can suggest it to my dad. What's making you concerned about his thyroid, and what needs to be ruled out? If I'm going to make a case for my dad, it's gotta be a good one. I can always win him over if I'm well-informed, but if I say I'm going on a hunch, or because someone on the internet told me so, he'll just ignore me.
Because he's 11 years of age, seems to be slowing down, has gagging and vomiting, etc....if bloodwork has already been done, that would give you an indication of his vital organ function..., but the thyroid function test should always be included, it's a seperate test but it is standard to advise the owners to test thyroid in older pets, particularly when symptoms are present that could be caused by disfunctional thyroid function. (the symptoms can be numerous, and those symptoms may or may not be present in all patients, bloodwork is the only way to determine)
If a pet lives to a ripe old age, and particularly those with limited activity, or those overweight, etc, more than likely they will develop arthritis at some point in time. One can assume arthritis should be checked for when the owner is very observant to their dog's behavior...slow to rise, difficulty walking/sitting/jogging in the yard, painful joints, decreased interest in activity, obvious signs of discomfort and pain. Xrays can help the vet evaluate this as well.
Again, if the vomiting and gagging continue or worsen, rule out esophageal problems, megaesophagus, tracheal infection/inflammation, GI ulcerations, bacterial infection somewhere, kennel cough, etc.
