That subnet certainly isn't normal, i'm not sure what is up with that. The correct subnet should be automatic and you should not need to add the 192 at the end, this makes no sense to me.
I would reset the router in either event, you should not need to add anything to the default subnet. Also, the static IP settings aren't meant to have your default gateway IP, they only work if you are using the correct range of IP's. Although static or DHCP shouldn't make a difference here.
It may help to see the ipconfig /all from when you are connected to the schools network.