Does Default Gateway IP = Router IP?

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.