I connect to the internet using a static ip given by my ISP. When I plug in the cable directly into my pc or my laptops, I'm able to join the LAN composed of the other computers in my neighborhood. But when I use my router to connect all my laptops and pc at a time, I'm not able to connect to the LAN anymore (I only see my laptop and pc in the internal network, not all the computers in my area). How could I connect to other machines of the extended LAN using a router (this is actually the WAN side of the router) ?
A router ignores traffic that's initiated from outside (internet) computers. This is because it uses Network Address Translation, a process that lets multiple local devices connect to internet using only one public IP address.
In short: If a local computer is surfing on internet, its local IP address (E.g. 192.168.0.x) will be translated to the public (ISP) IP address by the router. But if a computer on internet will access that public IP address, your router will ignore it.
Access One Computer Using Port Forwarding
You can set up port forwarding in your router. This way, a computer from internet that connects to your public IP address will be forwarded to the IP address of one computer you choose in your network. Warning: you should only do this for encrypted protocols such as SSH, but even then it can be dangerous. Don't do it at all for normal Windows networking.
Access the whole network using VPN
If you want access to your entire network from outside, you will need to install a VPN server on a computer inside your network. Some routers like the D-Link DIR-640L have an included VPN server, this will do the job too.
A typical wireless router usually prevents machines from the outside (WAN) to see the machines connected to the inside (LAN). Moreover it probably uses NAT (Network address translation) which prevents machine from the WAN to directly address the ones on the LAN. On the other hand this feature enables to share a single internet IP with multiple computers.
And because you have only 1 static IP address to connect to the Internet I don't how you could connect multiple machines directly on this network.
So a way I could think of to join the network with multiple machines is to check if your router as a mode of operation called "Bridge". In this mode the router will disable all network services like DHCP and NAT and connect directly all computers to the other network (almost like a simple switch). However this means that you would need to set static addresses to all you computers but like I said you may not have this possibility.
Another solution could also be to put one of your machine in the DMZ of your router. For this you need to connect your router exactly like you described. Like you said only the machines inside the router see themselves. Go in the DMZ tab in the configuration of your router and put the IP of one of your PC in as DMZ. This needs to be the IP given by your router, most likely something like 192.168.x.x. This machine in the DMZ should now be able to see all others outside of your router. This will not make all internal machine see the external ones but at least one will do and the others will be able to use the internet.