Can a cervical herniated disc in the neck cause slurred speech, difficulty talking, and swallowing? Sometimes choking on thin liquids?

Evaluation needed. Swallowing problems are most commonly related to neurological disease. The most common reason for these symptoms would be a stroke in the brain stem but other reasons need to be investigated for an appropriate treatment plan to be set.
No. Cervical discs classically cause radiation to the upper extremities.
No. Those functions (swallowing, talking, etc) are all mediated from above the c-spine. Those symptoms are worrisome for stroke and evaluation should be sought on an emergent basis.