Is it possible

So if I install it on my second computer, using my own steam account.. I should be able to play LAN as long as we have our internet/wifi button set to off?
 
Yes, an ethernet cable will do. Make sure you have the right one (straight, not crossed, if I'm correct).

OT, but in fact it's the other way round: a crossover cable is for connecting two computers directly. In real life it's largely irrelevant though, as any network hardware bought in the last decade or so will be auto-sensing and work with either wiring regardless of whether it's a direct connection or going via a switch.
 
So if I install it on my second computer, using my own steam account.. I should be able to play LAN as long as we have our internet/wifi button set to off?

Unless it recognizes, through the shared LAN account, that another computer is using it.

Short Answer: Haven't the foggiest idea.
 
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