So I picked up my car from Chris22104 this past weekend. Very nice guy. He knows these cars well and I was happy with the purchase. He showed me his hot air mod, which he needs being in Buffalo. For him it helps the engine heat up faster, and placing the sensor for auto stop in the airbox allows him to auto stop in sub freezing temps.
I almost did this myself, but being in MD, where we don't have sub-freezing temps but maybe 50 days a year, I question its need on my car. So I started looking around, and saw different people doing the mods differently. I couldn't discern whether the intent was solely for better below-freezing cold weather performance, or better mileage year round. The threads seemed to focus on the how, but I want to know the why. I have a few theories on how WAIs affect our cars, but want to discuss them to make sure I am catching all the important stuff. EDIT: I've had people PM me about theories, so I am editing the list. Perhaps with enough input on and offline, I can turn this first listing into the answer instead of just the question.
Theory 1 for how WAI helps -
Increased use of auto stop - As outlined above, this would be an advantage. But for my environment, I would only have gains on those sub-freezing days. Although getting the engien up to temp faster allows AS to start sooner.
Theory 2 for how WAI helps -
Heat engine up faster - An engine in it's proper temp range will run better for many reasons. If cold it will enrichen the fuel mixture, have more blowby, and have to churn cold oil.
Theory 3 for how WAI helps -
Less dense air takes less fuel - If the air is warmer, it's less dense. So in closed loop this will result in a rich mixture, which will then be corrected the O2 sensor by using less fuel until the engine runs at the mixture the ECU wants. In open loop, while the engine is warming up, the IAT sensor will sense warmer air, and will pair less fuel with it than if it were colder since it is less dense, and warmer air evaporates fuel better.
Theory 4 for how WAI helps -
Less dense air is easier to pump - not much explanation needed there.
Theory 5 for how WAI helps -
Less dense air makes less power, which means we need a larger throttle opening, which means less pumping losses. - It's kind of effectively decreasing the displacement of our engines. Larger throttle openings lead to better specific fuel consumption. AutoSpeed has a great article on BSFC complete with a BSFC chart for the Insight
here.
Theory 1 for how WAI hurts -
Less dense air makes less power - Is this good or bad?
Theory 2 for how WAI hurts -
Less lean burn or ignition advance use - Having higher air intake temps might lead to conditions more prone to knock, so the ECU would be less agressive with ignition timing, and perhaps not let the car into lean burn as often. Anyone notice a difference in lean burn use with and without a WAI? I've read stuff like "110F seems to be the peak for widening the leanburn window." in JoeCVT's thread about his automatic WAI. So apparently warmer air is better for leanburn.

I've tried searching for more on that, but without much luck. I'm really curious in what ways intake temps affect leanburn operation. Could someone comment here? I could just ask for clarification in JOECVT's thread, but that info would be so buried away at the end of a long thread that it might not do others any good in the end.
I'm now thinking that the advantages outweigh the disadvantages, and that this will help mileage even for those that don't deal with frigid temps on a regular basis. Please enlighten me.