This has a good chance of getting technical, I apologise in advance but there is no real way around that. The last paragraphs should have "simple" solutions but do read the rest as it might help sort things in your head.
You usually can get another router- you might just have to enter data into that where you would not the original (some ISPs do not need that though). Picking a random provider for this example you usually just have to find their FAQ page and add stuff like
http://www.draytek.com.au/faq/WAN/adsl.htm#ADSL_settings
Even if that is not possible then you can always add another router/switch/hub onto the existing setup- most people with cable modems end up doing this.
Secondly many ISP supplied routers come with their own firmware (there is a whole bunch of people who like to hack these to do many things) and looking around it seems this one comes fairly locked down.
Also you seem to be confusing two things
IPv4 as part of the spec has a few IP ranges that are not used on the public internet (
http://www.inetdaemon.com/tutorials/intern...addresses.shtml )- a secondary protocol called DCHP can be used to assign addresses to devices on a network without them having to ask for their own special ones (and you having to sort it out which is nice if you do not want to spend 5 hours configuring a big network and keeping things up to date)- this using "special" ones is known as using a static IP address.
The latter part of what you just described is mainly for people that want to run servers or handsets for voip on the network and need to forward static IP addresses to the devices without the chances of a collision with something the DCHP protocol assigns (DCHP can be told to only give out addresses in a certain range leaving others unused/free).
NAT is a method by which one IP address (you router and depending on your ISP maybe a few things there as well) can be used to serve multiple devices- you change the port numbers and as the port changed to is known the data goes where it is needed and that can mess up some protocols or implementations of them (xbox live often providing a good example of this).
What you actually want to do here is either set up port forwarding, tweak firewall rules (your router often has a basic firewall built in) or more likely set up a network DMZ (demilitarized zone). The only reason for restricting the DCHP pool here is as part of security setups (if you have just set up port forwarding you can make it only go to devices that know about it- those that just snatch up DCHP addresses like junk virus ridden laptops and phones that randomly connect then have less chance of compromising network security or vice versa you can tell certain ranges that they can not use the internet just the internal network- big companies can use more exotic versions of this to restrict things by department).
Short version- if it does not work out of the box then you have three options
1) Port forwarding. Go into the network settings of the 360 and tell it to use a static IP address (you can stick it in the DCHP range if you want- just pick a higher number as it is unlikely your home network will ever be rocking 90 devices at the same time), next go the router and choose port forwarding and point it at your 360.
More modern routers and network setups can tie a device (usually a mac address) to rules and forwarding saving you having to mess with an IP address but the ISP supplied garbage rarely has such niceties.
2) Set up a DMZ- this is mainly an open access portion of the network the router will say nothing about and allow to carry on regardless (give or take some other options/overrides). Make sure the live protocols and port ranges are
3) Check your firewall is not blocking Live. Similar to the DMZ the firewall may be configured to only allow certain traffic out of the network. Many devices will already have live plugged in.
