Confusion in IP address configuration! Traceroute/pathping

When I query for my public IP address it obviously shows an IP address from Public cass typically 1.22.112.x. I have a Dlink router which is configured to DHCP and I am guessing my router also gets inteet edge IP from my ISP's DHCP server.

The webinterface of my router shows following configuration (for inteet); IP : 10.25.x.x subnet mask : 255.255.240.0

My first confusion is why the router gets an IP from again Private class? How my router is able to route this private IP as they are not meant to be?

When I pathping google's DNS server this is the output i get

C:Usersblahblah>pathping 8.8.8.8

Tracing route to google-public-dns-a.google.com [8.8.8.8]
over a maximum of 30 hops:
  0  Blahblah [192.168.0.102]
  1  192.168.0.1
  2  113.193.240.17
  3  115.111.205.13.static-mumbai.vsnl.net.in [115.111.205.13]
  4  172.31.33.189
  5  115.113.165.98.static-mumbai.vsnl.net.in [115.113.165.98]
  6  209.85.142.167
  7  216.239.41.155
  8  google-public-dns-a.google.com [8.8.8.8]

Why i dont get an entry from 10.25.x.x series? My request should go from my router to the ISPs router and then to the inteet. It should have an entry of ISPs default gateway(10.25.x.1).

Lastly when we tracert/pathing an IP address lets say facebook/google, there's also no entry from network 1.x.x.x. I typically get public IP from this series so should at least get an entry from this network as well.

FYI My ISP Tikona Digital Networks has a centralized system where we're given an useame and password. A user is required to login every time in order to access the inteet. Secondly, all of the IP addresses are dynamic.

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