If I had thyroid cancer, would I be able to tell?

Depends. The common types of thyroid cancer are slow growing and are typically not associated with physical symptoms. Unless the cancer is in a big thyroid mass (> 2 cm), you might not feel anything. The best way to evaluate for thyroid cancer is to have a thyroid ultrasound so you can see the gland visually. If abnormalities are seen on this exam, a thyroid blood test and thyroid uptake & scan is needed.
Not usually. Usually the only way to know is by biopsying a thyroid nodule, usually with an ultrasound guided fna (fine needle aspiration).