Does Default Gateway IP = Router IP?
Solution 1:
The "default gateway" is simply the IP of the next hop for traffic that is not explicitly routed anywhere else. This will, almost certainly, be an IP address on an interface of a local router. However, it may not be an address on which you can manage the router.
If this is for a "home-type network", the inside IP (your default gateway) and the management address will probably be one and the same, but in a "large/corporate-type network" I wouldn't be so sure.
Solution 2:
I'm going to hit you with the big "Depends": on some networks, the default gateway maybe a Layer 3 switch; you may also find that the default gateway (typically the LAN IP) is not the same interface that the management console/Web GUI listens on.
If you're talking about SOHO/SMB "flat" networks, then yes, the majority of the time the default gateway will also be the router's IP.
Solution 3:
To get the default gateway parse the output of the "route print -4" command.