A high TSH usually means the dose of synthroid is not enough -- at 7 months, the TSH normal range is about 1.5-2x that of adults -- so normal may be up to 8.0 according to some research.
Under-treated hypothyroidism can cause some delay in development and growth -- please make sure you are working with a pediatric endocrinologist who can closely monitor the replacement and make sure it is adequate for your son.
[This was weird, I posted again but it didn't show up]
If you happen to have your son's thyroid labs & ranges, post them here for the doctor-as they may be of help. As well, the medication and amount he is taking.
This will not answer your questions and am glad you are asking somebody who can give you good direction.
That said, as a parent, I understand your concerns. It's especially difficult to understand what's going on because they are too young to convey how they are feeling.
Anyway, given your circumstances, I think I would hunt around for a good infant/toddler blood pressure monitor with heart rate. You need the correct cuff size.Perhaps there is a battery operated one-just like those for adults?
I would speak to a health professional about one and learn from them what is considered a normal B.P/Heart Rate for age brackets. While of course, letting them know of your son's condition.
I do know infants/toddlers have higher heart rate and then as they age (such as my young teen daughter) their B.P. is usually lower than adults.
I think I would be checking it periodically and then check it when he isn't feeling well and acting unusual.
Best to you and your son, ttatina.