A quick install of the Creekstone IPAdvisor
(
http://www.creekstone.net/software/ipadvisor) program will do that.
It will e-mail you a notification when the IP address changes (renews or
looses its lease) as well as give you a web page link to a history of your
different IP addresses, including the current address. I use it because my
cable modem ISP provides a dynamic dhcp assigned ip address and I want
remote access to telnet, ftp, and web services on my home Linux server from
work. I don't use it for a full blown webserver since I'm just hosting a
few peprsonal HTML docs but I suppose you could even host a full blown
website provided your ISP doesn't monitor for heavy http traffic from the
Internet.
As far as the router is concerned I think it will work from behind almost
any router, firewall, etc. A friend of mine uses it for his Bellsouth PPP
DSL connection from behind a Linux router/firewall configured with NAT.
Hope that helps.