Assign a different IP address to eth0 on each of the machines, say 192.168.1.2 and 192.168.1.3
Connect the two network cards together with a cross over ethernet cable, or, if your router has multiple LAN ports, plug each computer network card into the LAN ports on the router with a standard ethernet cable.
Not really if you're configuring your network statically.
No effective difference.
Nope. You asked about a local area network, this will let the two communicate with no internet access and does no routing.
If you just want to share your internet connection, a residential wireless router will do the job with less bother than trying to make one computer into a router. Plus you won't need to have one computer on 24/7.
Mghis, if you want to share your Internet, you are need those commands
echo 1 >/proc/sys/net/ipv4/ip_forward - allow forward
iptables -t nat -A POSTROUTING -o eth0 -s 192.168.0.0/24 -j MASQUERADE
eth0 - Internet interface
192.168.0.0/24 - your local network