1 . The result of checking a capacitor for continuity could vary depending on the value of the capacitor. A very small value capacitor will generally show no continuity, (open circuit).
A large value capacitor may show a brief continuity (short circuit) as the capacitor charges, then will show no continuity after a short time, (open).
If a large capacitor were already charged (previously tested) a later continuity test may show an immediate open circuit. If the capacitor were discharged then retested the result should be as above.
After a continuity test on a large value capacitor a volt meter could be connected to show a residual voltage across the capacitor pins.
2 . If there were an internal ESD component it would cause a short at some high voltage level, (for example during a fast electro-static discharge event). Most ESD components are capable of discharging a fairly large current for a very brief pulse time, (on the order of amperes not milli-amperes). If a test voltage were to be raised slowly to a high value it may damage an ESD component if a large current were allowed to flow for more than a brief amount of time. (But before that would happen it is more likely that any other connected components would be severely damaged).