During a purported oral surgery, someone pushed up the back of my mouth with an instrument, until the bone cracked. Could he/she have damaged the brain stem?

Unlikely. Since you had the ability to post this question. The brain stem controls many basic functions such as breathing, so you are okay.
Not likely. The brain stem is quite a distance from the oral cavity, in medical/surgical terms. Also, the brain stem is in the center of the vertebrae and is quite well protected by them. During tooth removal or other types of oral surgery, fracture of the alveolar process (the bone that surrounds teeth) is very common. Sometimes this bone is removed surgically, to enable the removal of a tooth.
No. Check image, palate manipulation is within pharynx, in front of brainstem & spinal cord. The posterior portion of the palate is soft / cartilaginous, not true bone.