|Routers connect networks. In home applications they are usually used to connect the a public (the Internet) with a private network (192.168.x.x). Since there is no way to talk to a server on the Internet then there are no packets ever made with IPs outside of your private network; unless you specify a public address in a ping.|
If you try to send a packet to a private IP that is not on your network then the packets get dropped by the first Switch. If you use a Hub then they get sent to every one and the NIC will drop them unless you have a sniffer setup.
If you try to send a packet to a public address that has turned off their router then the packets are dropped at the edge router of your network. You can over run routers this way and this is sometimes used in DDOS attacks but most routers have built in software that will prevent this.
Not sure if this answers your question.
Most of the time we will ask you to ping your router because most home owner buy Router/Switch/Wireless Access Point combination devices. They are three devices made into one. If you can ping the router then every thing is working.
If you have a Router hooked into a switch Hooked into a Wireless Access Point (WAP) then each of these devices may have an IP for hosting a configuration page. (In the old days you had to telnet tot them) In this case you may be connected to the the WAP but not see the router. When trouble shooting you can not use TRACERT because these are layer 2 devices but there is really nothing to them. Either the lights on the port light up or they don't
I am not sure what issues you are having, but some more detail might be needed in your question.