questions on powering a router with 9v alkaline battery
I'd suggest your battery terminal voltage dropped, which would heat the battery due to it's internal resistance. There is no way of knowing exactly what happened in your router since we can't see the schematic. 

There is no reason that using a 12 V battery (however, and with whatever cells you choose) won't work...BUT it has to be able to deliver the current needs of the router and have enough mA/h capacity to suit your expected runtime. 

I'd measure the current into the router when active and using your AC power supply. Then you have some starting point to configure a battery solution. 

Most AC plug pack equipment is tolerant of around 10% or so variation. It may be that you could use  9 * Alkaline C batteries (13.5 VDC for new batteries), but if you are really drawing anywhere near 500 mA then the runtime to 12 V (or some lower voltage, say 11.2 V) is likely to be only a 4-6 hours since you can only discharge the cells to about 1.2 V at best.