Is it safe for a 3 month old to be on Synthroid (thyroxine) for low thyroids?

Absolutely. It is essential for thyroid hormone to be replaced if an infant is low in hormone. There is sooooooooooo much that is going on with regard to growth and development in those early months that is affected by thyroid hormone. Lack of the hormone can lead to very poor growth and feeding, developmental problems and long term intellectual problems. Give her the best long term outcome and give her med.
Yes. This replaces what the gland should be making. If you don't want your baby developmentally handicapped you need to give it as prescribed and do the follow up blood tests that monitor effectiveness on schedule.