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A URL slug should be short, readable and made of real words. Search engines read hyphens as word separators, and a clean slug tells both people and crawlers what the page is about before they open it. Removing stop words usually makes the URL stronger, not weaker.
Usually yes. Words like a, the and of add length without adding meaning, and shorter URLs perform better.
Hyphens. Google treats them as spaces; underscores join words together.
You can, but set up a 301 redirect from the old URL or you lose the rankings and any links pointing at it.