Which part of the brainstem connects the brain to the spinal cord and controls many automatic functions such as breathing and swallowing?

The medulla oblongata is the lowest part of the brainstem, and it connects the brain to the spinal cord. It is responsible for controlling many automatic functions of the body, including breathing, heart rate, blood pressure, and swallowing.

The medulla oblongata contains several important centres, including the respiratory centre, which controls breathing, and the cardiovascular centre, which regulates heart rate and blood pressure. It also contains centres that control swallowing, vomiting, coughing, and sneezing.

Hence, the correct option is medulla oblongata.