What are the odds that if I have hypothyroidism that I will end up passing it on to my children?

Risk is higher. Thyroid disease is so very common in the general population of the U.S. If you have hashimoto's disease then the likelihood of your child getting thyroid disease is somewhat higher than that of the general population because there may be a genetic susceptibility which isn't well understood.
Depends on Type. Hypothyroidism has several possible causes that influence the answer. Isolated disease due to auto immine thyroidits does occur in family clusters and regular testing can identify kids early if appropriate. Isolated cases occur in regions of iodine deficiency that iodinized salt will avoid. A recessive metabolic form is unlikely passed to direct offspring. Check with your doc to identify the risk.
Depends on Type. Really depends on the cause of thyroid disease for example if you have an autoimmune disorder your children are at higher risk of developing something similar.