To begin with the answer to your question - goiter is a life time disease in some but not all!!
Goiter is an enlargement of the thyroid gland usually due to the effect of a hormone called TSH. The cause for high TSH may be excessive secretion from the pituitary gland tumour or else where but usually is due to inadequate function of the thyroid gland to meet the demand for thyroxine.
This demand may go up in puberty (adolescence), pregnancy and certain disease conditions. The goiters which come up in such instances usually resolve at the end of such a demanding period. Also goiters formed due to lack of hormone production by the gland can be corrected at initial stages (if recognized) by replacement of thyroxine.
Once a goiter develops, again the size can be reduced by high dose thyroxine and iodine therapy. Ablation of the gland with radio active iodine is also an option.
Finally, if the goiter is too problematic, surgery becomes an option.
To answer your second question - not the goiter but low or high thyroxine can cause enlargement of the heart in many different mechanisms. (dilated cariomyopathy in hypothyroidism, high output cardiac failure in hyperthyroidism) - for more information on enlargement of the heart you can go through my hubs on heart failure.