if you have configured your home router to Cloudflare's or Google's
DNS settings - now, does it make any sense to alter you DNS settings
-again- at the your Laptop or Mobile devices?
If your laptop and mobile devices go 'off' your network and connect to wireless networks at other locations (at school, at the doctor's office, at a restaurant) then it makes sense to have configured those settings at the device level.
Can DNS settings be configured twice i.e both at router level and
They can, although generally, once settings are configured at the device level, router level settings will be ignored.
Does it provide any extra benefit - like more privacy or security?
It's good to enable device level settings in case the device travels. It's good to enable router level settings in case a device that doesn't have local settings comes onto the network. So, yes, there's extra benefit to doing both, although each device will only benefit from one level of setting.
What happens if you configure you router to Cloudflare's DNS settings
while your device is configured to Google's or Quad9's (or any other)
DNS settings? Which DNS resolver is in action in such a scenario?
Again, only one level of setting will take effect, the most local setting (device before router). If device resolver is configured, that will take effect, and router setting will be ignored.
Does such a configuration double the online privacy and security?
There's no doubling up on a single device, but as mentioned, if the network contains both devices with device level settings and devices without device level settings, then having the router level settings provides a safety net for otherwise naive devices.