LRP? Get a $25 used (of course) 486 or whatever. Once it's setup and running, you don't need the monitor or mouse. BIOS might require the keyboard. Don't need a hard drive either.
What would I install the linux software on without the hard drive?
There are linux setups that you can run off a bootable cd-rom, hence no hard drive needed.
In order to use the laptop as a router you will need two rj45 network ports. Since you only have one slot for a card and, I am assuming, no onboard nic you're out of luck as far as I know.
I know it's a bit late, but maybe usefull for other people. I think that if you want you can use your only one ethernet notebook as router simulating two nics this way (on Linux. If you are on Windows not sure if you can):
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