No, B12 shots have nothing to do with thyroid issues; however, you should get both thyroid and B12 levels tested. Many people who have thyroid problems are deficient in B12.
No but it wouldn't hurt!
Seriously though, many people who have underactive thyroids have a lack of B12 in their diet, especially vegetarians. ( I should know!)
Have you had any blood tests to know that you have an underactive thyroid?