Hi Lisa,
Thyroid medicine is one of the lower costing and longer tested meds around... for humans and dogs. It's something that will cause a dog no harm to take for the rest of its life, provided that it doesn't cause the levels to rise above normal.
Samm was put on the medicine last October due to borderline levels that were declining in the last several years. The dr. told me what to look for if the level became too much. Basically, she would start roaming around, panting more, unable to settle down.... similar to a person overstimulated on caffeine.
She's been fine. Her hair is thicker than it was (not that it ever did thin out)... gleams, her muscle tone is excellent, and she regained her lost 'spark'.
We have a scheduled appointment for Feb. for the dr. to retest her thyroid level (as directed by him) and probably will decide to continue the medicine.
This is a base guide for what your vet should do...
As for with puppies, I don't know what the medicine will do to them. If things are left the way they are, she is likely to reabsorb them again. So whatever the dr decides might be the best option.
good luck,
toodles