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There are several different types of Thyroid diseases which would cause
a hypothyroid condition (reduction in the output of the thyroid, mainly
thyroxin). Except for ones caused by infections, the treatment is
generally thyroxin pills. Hypothyroid conditions caused by infections
usually disappear when the infection does...this doesn't sound like the
case with your wife.
Thyroxin orally does "shut down the thyroid" through a feedback loop
involving the pituitary (I believe). The pituitary "thinks" that the
correct amount of thyroxin is being produced so it doesn't have to
tell the thyroid to produce more. This process is reversable! I have
Hashimoto's thyroiditis (an autoimmune condition) and was on thyroxin
for approx 6 mo when my endocrinologist suggested I not take the pills
for 6 wks. When I was retested for thyroxin levels, they were normal.
I still get tested every 6mo because the condition might reappear.
The pills are safe and have very few side-affects (& those mostly at
beginning of treatment). Having a baby might be a problem and would at
least require closer monitoring of hormone levels.
Thyroxin controls energy production which explains sleepiness, coldness,
and weight gain. There is also water retention (possibly around heart),
changes in vision, and coarser hair and skin among other things.

I am not a doctor, so I'm sure I mistated something, but the important
thing is that thyroid problems are usually easily corrected and if they
aren't corrected can cause problems in the rest of the body. Get a
second opinion from a good endocrinologist and have him/her explain
things in detail to you and your wife.