What are the actual specs of the machine?
If you're just serving files and don't need any graphics, the best method is to do a command line only install. I usually use Debian because you can install a base install of like 400 MB and then build whatever you want on top of it. I even do this on my desktop machines that I add X Windows to. Kubuntu is based on Debian. Here's a good Debian tutorial
on how to setup a box via the command line.
I haven't messed with wireless enough to know #2.
From your post, I would guess it will be easy enough for you to get the hang of this because 1) you don't seem afraid of learning something new, and 2) you have a clear objective that you'd like to achieve. The latter will make recognizing your problems easier because you know what you're trying to accomplish.
Good luck and I hope you give linux a try!
EDIT: Damn Small is probably more limited than what you want to do. As a newbie, it'll be easier for you to work with pre-compiled packages. DSL has a limited number of packages available for it. I would go with Debian Lenny (will be stable soon) or a Ubuntu "Alternative" install which will let you get a stripped down Ubuntu. Debian has like 20,000 packages in the official repositories, plus unofficial repositories like debian Multimedia...