Is there typically an adjustment period to hypothyroid medication?

Yes. Your body will adjust to new meds and you may have to increase or decrease the dose over time. Go by how you feel and not just lab numbers.
Yes. The most common cause is autoimmune or hashimoto's hypothyroidism. Especially in the beginning, this can fluctuate leading to unpredictable changes in the hormone level. This will require monitoring and dose adjustment.