Is the screen broken?

Those black spots are either caused by the display going faulty, plugging or unplugging the display while the battery is still connected or too much pressure was applied against the display when it was being removed for the first time.

Edit: I forgot to mention try reseating the screen cables first it may just be the display cable not being fully seated onto the connector.

Either a new LCD display will fix this otherwise the backlight circuit is damaged and needs to be repaired which happens when the battery is still connected and the display cable is plugged in.

If to does end up to be a backlight circuit fault the iPad will need to be taken to a reputable repair store that does microsoldering.

I would probably leave it at that as personally I wouldn't spend too much money on a 1st gen iPad mini.