Do the water droplets that splatter during ablution (wudu') or when spraying a car, and which enter a fasting person's mouth and are swallowed with saliva, break the fast? And if so, must the fasting person close his mouth to avoid them?

A fasting person must be wary of anything that might invalidate their fast, whether it is a small or large amount. So, if raindrops fall into their mouth and they swallow them intentionally, despite being able to spit them out, their fast is invalidated. However, they are not obligated to close their mouth because that entails hardship. They can expel the water and not swallow it, just as in rinsing the mouth.