So random IP scanning is harmless unless your PC has a static public IP address.
I agree that in cases where you have a good firewall (in your example a router which are increasingly common) security is less of an issue. Provided you've changed the default password and/or checked the remote management settings that is.
Most routers now will not allow configuration from the WAN unless remote management options are enabled, but this isn't always the case - especially on an older router. If it's more than a few years old you may also have left the default Telnet admin options switched on which was a common form of hacking a few years ago and there are still people automatically sweeping the IP range of the net looking for telnet connections.
Routers are increasingly configured by default with a home user in mind - but dont trust on this. We're all technical people here so be sure to use your knowledge and check the remote admin options / change the password.
A DHCP or static IP address on the router is neither here nor there though.
Dont forget also that this is an international forum, and when you think that even parts of Britain and America still dont have broadband (my brothers house has no internet at all), it's not hard to see that when giving advise on security it's dangerous to be dismissive based upon an ideal operating environment.
Remember also that a router can be hacked by some of the clever ones (thankfully there are few). If the US military can be hacked and have all their data downloaded, if Microsoft can be hacked and have the Windows source code downloaded, then i'm pretty sure the rest of us dont really stand a chance...