All Zip / DMG files state 'is damaged and can't be opened'

You shouldn't have to use either Stuffit or Unarchiver to open most downloads. Apple provides it's own app called diskimage mounter.

Let's test in a new user or the guest user to eliminate problems in your User's folder.

CREATE A NEW USER or use the Guest User.

Go to System Preferences --> Create a New User in Users & Groups.

Switch to the New User by logging out under the Apple in the Menu Bar.

Log into the new User.

You can skip logging in with an Apple ID. Hit continue then skip.

Only Apple apps will be in the Dock.

Open Safari and download any file.

Double click to mount.

Do you still see the issue?

If no, then the problem is in your User's folder. Download EtreCheck. Etrecheck is a diagnostic tool recommended by Apple Support personnel. Paste the results in a reply. This is similar to Apple's System Profiler but has more info that can diagnosis issues on your Mac.

http://www.etresoft.com/etrecheck

More Info on Etrecheck

https://discussions.apple.com/docs/DOC-11591

You do qualify for free support from Apple since you have a new Mac. Time requirements for purchasing Apple Care has changed.

Generally you get 60 days from the purchase date to buy AppleCare. Apple has a page where you can check your exact device’s eligibility. https://checkcoverage.apple.com

No matter when you buy your AppleCare+ plan, the coverage starts from the day you bought your device. So you don’t get an “extra” two months of coverage if you wait two months to buy the plan.

See some articles Download items from the web - Apple Support

If you can’t download an item from the web - Apple Support

If a plug-in slows Safari performance - Apple Support