Can thyroid adenoma develop into carcinoma?

No. A true thyroid adenoma is a benign tumor. However, this can only be accurately assessed once it has been removed surgically -- not by fine needle aspiration. There are molecular alterations seen in carcinoma than are not seen in adenoma. However, some of the molecular changes are cumulative, like p53 mutations -- which are expressed in higher frequency in cancer.
Maybe. If you have a thyroid nodule that has "follicular cells" or has been labeled a "follicular neoplasm" by fna, you can't tell whether or not it is a carcinoma or adenoma. The only way to know for sure is to remove it with surgery. Certain findings on ultrasound and increasing size of a nodule make it more concerning for a carcinoma.
Thyroid. The only way to differentiate between a follicular adenoma and follicular carcinoma of the thyroid is to remove the nodule surgically and carefully examine the periphery of the nodule for signs of invasion into/through the capsule or to find invasion of blood vessels/lymphatic spaces.