Do refurbished phones have new screens?

This is a very vague question. If the device needed a new screen then it would have been replaced. If the screen was ok but the glass was scratched they might replace that. I do not believe apple sells refurbished devices that look used. Apple claims "Guaranteed Apple Quality" when purchasing a refurbished device from them.

http://store.apple.com/us/browse/home/specialdeals/refurbfaq_popup

If you buy an item from the Apple online store that says "refurbished", it will come repackaged with all the items that a new device comes with, and it will be eligible for Apple care + (within 60 days of purchase).

If you are issued a replacement device (IE: You submit a device for warranty service), you receive only the device, no accessories.

The refurbished devices are remanufactured. New Screen, outer shell, buttons, and battery.

Hope this helps.

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is repackaged (including appropriate manuals, cables, new boxes, etc.)."

I believe so.