i have had foot pain for years, and hypothyroid, but no clue if they're related! my feet get sore quickly when i'm standing or walking, and walking gives me arch pain easily. i also injured my plantar's fascia (the tissue that supports the arch) while running, and it's taken a couple years for the pain to fade.
fluid retention can cause puffy, swollen feet, and put pressure on tendons. so keep drinking your water. i feel good with 1-2 liters per day, some people advocate more like 2-4 liters per day. pain in the morning is also a sign of fluid retention in your tissues (builds up over night when you're lying down).
two other things that have helped my arch pain have been losing significant weight, and wearing shoes with a modest heel - 1-2 inches, as opposed to flats. flats kill my feet!
i don't know much about thyroid and menopause, other than that many women only develop thyroid problems at menopause, so i would just guess that if you already had known thyroid problems they could shift at menopause.
hope your feet are feeling better soon! i would mention it to your doctor when you go to adjust your thyroid meds. he/she may want to keep an eye on it, especially since it seems to have developed suddenly??