1) adding the network card
For your network card to function properly you need to have the right module in your kernel. To find the correct module you should use lspci and make not of a new Ethernet controller entry there. Then you can either google for it or go to
http://www.linuxquestions.org/hcl/index.php an see what is there.
Once you know the modules you can try to modprobe module_name it to see it everything is ok. If it is ok you should add the module_name to your /etc/rc.d/rc.modules so it is loaded when you boot you machine(Assuming it is too slackware).
After that you need to configure the IP address and routing between your computers. I think there is a network configuration utility named netconfig. if there isn't check to see if you have a /etc/network/interfaces and add there the necessary information so the system can configure your network when it boots. If you don't have that file you can do it by hand using the ifconfig and route command.
/sbin/ifconfig eth0 192.168.0.1 up
sets up your network card IP address to 192.168.0.1
/sbin/route add default gw 192.168.0.2
makes the system send all its messages to that host so that they can be forwarded. This is useful as you probably want something like
Internet <---> Computer with 2 cards <---> Other computer
you can then forward messages using iptables.
2) You have to consider how big are the transfers going to be and how much security you want.
The fastest way to setup the system is just to use ssh and scp. However the encryption will take some bandwidth away. Just put a ssh server in each machine and you will be able to copy files back and forth.
3) The security you need depends on the thread level. For slackware you should disable the inetd so it doesn't start uneeded services and change the /etc/ssh/sshd_config to say "AllowRootLogin no" so that people aren't able to crack your root password remotely. If you need added local security you should password protect your bios and make the bios ask for a password every time.~
So theorically you should only see a server in port 22 when you do netstat -ap