To answer your heading question: Towing adds more strain to the engine creating more fuel consumption and oil use.Mikered30 said what I was going to and his assumption on the use of a cooler IS to save the tranny from over-heating.
Unless you can locate the previous owner and get some background info on his towing habits, etc. I would steer clear of this one.
I have an '04 Matrix and you'll never see me towing anything with it as I don't think they are strong enough. That's JMHO though.
I have an'02 Tahoe that does all my towing needs. (21' travel trailer)