My Endo told me when starting out on Synthroid and also each time he raises my dose that I may have some palpitations or "heart flutters." Each person is different. I have noticed them a bit, but nothing terrible. He said it should go away within a week or two after in your case, starting the med...in mine, when he raises it. Nothing to be alarmed about.
I totally agree with goolaraa.
I have had no side effects from my T4 medication.
Hashi's is an autoimmune thyroid disease. When you have Hashi's, your immune system is producing antibodies that attack the thyroid, eventually causing hypothyroidism.
"Thydroid deficiency" is also hypothyroidism, but from a cause other than autoimmune or an indeterminate cause.
There can be side effects from levothyroxine. By and large, it's well tolerated since it's not a medicine, per se, but a replacement of the hormone your thyroid would produce if it could.