I am aware that most people with narrow palates have smaller mouth width. however, is it correct that people with wider palates automatically implies that they have wider mouth widths compared to people whose palates are narrower?

Not necessarily. There are no automatics, just comparisons to averages. Not sure how this really matters.

Yes. And as the roof of the mouth is the floor of the nose, a wider palate indicates better upper airway volume, less snoring, chance of impacted canines in 11-12 year olds, better incisor function, and so forth.This is why ALL children should have their first (usually free) Orthodontic examination by their 7th b'day.