When your system discovers that your computer can reach 8.8.8.8 through your router on 192.168.44.1, it adds the 0.0.0.0 entry for you. It will delete this entry when it fails, or when 192.168.44.1 disconnects (hardware) or disappears (Software: ARP packets receive no response).
Complete nonsense. Check in any LAN with the switch or the router. There are stations on the L3 switch or the router will have 0.0.0.0 as the default gateway. So again, all this function checks is that there is a default gateway and it is alive. With the Internet it is connected only if this gateway goes to the Internet, but this check is certainly not here. At home, this is usually the case, but as I said, a couple of stations and one router to the Internet provider - this is not the only possible network configuration.