Does the cookie banner stop Google Analytics or other scripts until a shopper accepts?

No, not at the moment. The banner shows your message and remembers each visitor's choice (Accept or Decline) in their browser for a year, so they aren't asked again. It doesn't switch other scripts on or off based on that choice.

So if you add tracking or chat code (for example with the Code Snippets plugin), it loads whether the shopper accepts or declines. If your privacy rules mean a script must wait for consent, check with that script's provider how to set that up. Also make sure your Banner message and Privacy policy link describe the cookies your store really uses.