My router connected to 2 ISP and both run BGP. Received full routing table.
I've check the routing bgp adverstise and receive and seems to me ok as i can see all the routing table listed. For example not able to ping to the websites from the pc, when ping i can get the ip of the web(22.214.171.124) and also not able to ping using the ip address.
In your first ping, the output indicates (a.b.c.d) does not have a route to that destination. So it returns the icmp net unreach. The second ping simply times out. So, we do not know. Possibly try a traceroute to see how far you get.
In the third ping, it appears the namserver that router is configured for is getting a AAAA record for yahoo (ipv6). So, it tries to reach that first.
None of your output indicates a dns problem. The first one points to a routing problem on a downstream device (a.b.c.d).