Is there a way to monitor my router's network traffic, without having to replace the standard firmware? I want to be able to see each network host's network activity, but I don't want to replace the default firmware because of the risk of bricking the router.
With the default firmware, there is no good way to measure bandwidth on a WRT54G router. It is a consumer based router and lacks the features of higher end routers. Sorry for the bad news.
I know you said you do not want to but the best way is to flash the firmware of the router. It is not hard. Just read through all the documents before starting. dd-wrt has pretty docs.