The northernmost station:

This is the station before the terminus at Baramulla. If one looks at the map carefully, it can be seen that it is further north.
In case the proposed line from Baramulla to Kupwara is built in the future, the latter will become the northernmost station. There is also a proposed line to Leh from Himachal Pradesh, but Leh is around the same latitude as Srinagar.
Now, the Kashmir valley line from Banihal to Baramulla is not connected to the rest of the IR network (although this gap can be bridged in a 4-hour road journey from Udhampur to Banihal). The northernmost station on the IR network used to be Udhampur, which is now superseded by

These lines were always BG.
Now we move east. The easternmost station presently served by passenger trains is

However, goods trains run further east for a few km to Tirap Siding where coal is loaded. Although I could not find a good picture with this sign, there is this video of a road trip along this route with plenty of coal wagons:

Still further ahead is this now defunct station which was functioning from the late 1950s to the 1990s, when the MG line was converted only up to Tirap Siding as it was not considered worthwhile to extend the BG line here:

Lekhapani was thus the easternmost point of the Indian Railways, but not now. However, it is now planned to reopen this station after the last stretch of a few km is converted to BG.
This plaque can be seen near Lekhapani station:

Still further east are the Tipong colliery railways (2’0″ NG) which are NOT and never were part of IR, though we will take a quick look at them here:
These colliery lines have some B class locos which were earlier on the Darjeeling line. There are several longer videos of these lines on Youtube.
The main line even features in the 1972 film “Ye Gulistan Hamara”. If you are really interested you can see the film on Youtube, though the trains appear only for a couple of minutes. If you like typical Bollywood films of the 1970s and are fans of Dev Anand and Sharmila Tagore, you might as well see it.
A small 2’0″ industrial line was earlier functioning at a plywood factory at Namsai in Arunachal Pradesh, though this was also nothing to do with IR.
A new line from a point near Makum to Parshuram Kund has been proposed. When completed, this will be the easternmost station much further east than the collieries at Ledo.
Now to the south. That is easy enough. This line was built with BG.

And for the west, there is

Like Sopore, it is not a terminus but is further west than the larger station of Dwarka and the terminus of Okha. Dwarka is the westernmost station of some importance.
This line was MG and was converted to BG around 1980.
Varvala had this status for a long time. Then the Bhuj-Naliya MG line was built and Naliya became the westernmost station (with a lateral distance of about 10 km). The line from Bhuj to Naliya was closed for several years and now has been taken up for conversion to BG. When the line is completed, Naliya will regain the position-although there are plans to extend the line further west. No picture of Naliya station is seen on the net, so here is one of the next station Naliya Cantt, which is adjacent to the IAF base:
(Google Maps needs to be corrected as Naliya Cantt station is marked as Naliya station.)

Next we will see what the extreme points were in the 1970s.