The speaker actually claims it works by "interstitial suspension". I think he is referring to capillary action, which you are aware is involved here. He does not claim this method is faster, only that it is more environment-friendly. Shaking before wiping is a crucial factor.
Folding has 2 advantages : it reduces excess towel at the edges which is not used, and it makes the towel thicker so that the paper between the hands does not become saturated, when it will have no further drying capacity.
Yes, folding twice might be even more effective, depending on the size, thickness and quality of the towel. However, excessive folding leaves paper in the folds usused (ie dry) while the outer surfaces are overused and become saturated. So a single fold might be optimal.
Why not do some personal research and report your findings? (Be quick, or Floris will beat you to it!)
Yes. Consider throwing a ball at a bat which is held stationary: the ball is momentarily stationary but at all other times it is moving faster than the bat.
Now consider sweeping the bat towards an initially stationary ball: if the ball is not to stick to the bat, then it must be moving faster than it when it loses contact with it. (This case is identical to the one above with a different choice of reference frame of course.)
In neither of these cases have I taken proper account of conservation of momentum: the bat must change velocity slightly when it imparts momentum to the ball, so you can't hold it stationary or sweep it at a constant velocity in fact. But this change in velocity of the bat can be made as small as you like by making $m_\text{bat}/m_\text{ball}$ large enough so the argument remains true.
Why we can ignore the person holding the bat
In the comments there has been some discussion about whether the person holding the bat makes a substantial difference. They don't: they certainly can make a difference in detail and obviously are responsible for getting the bat into the right position, but their contribution to the change in velocity of the ball is small. To see this I'll take some numbers from this page (mentioned in the comments).
The ball has a mass of $m = 0.145\,\mathrm{kg}$ and its change in speed $\Delta v \approx 200\,\mathrm{mph}$ or $\Delta v \approx 90\,\mathrm{ms^{-1}}$. This means that the impulse delivered to the ball is
$$I\approx 13\,\mathrm{Ns}$$
Now, let's assume that the person holding the bat exerts a force equivalent to their whole mass on it (they can't do this for any length of time, and in fact they can't do it at all realistically, so this is a safe upper bound). If their mass is $100\,\mathrm{kg}$, then the force they are exerting is $100\,\mathrm{kg}\times 9.8\,\mathrm{ms^{-2}}\approx 981\,\mathrm{N}$. The ball is in contact with the bat for $7\times 10^{-4}\,\mathrm{s}$ ($0.7\,\mathrm{ms}$), so the impulse from the person holding the bat delivered during the collision is
$$\begin{align}
I_h &\approx 981\,\mathrm{N}\times 7\times 10^{-4}\,\mathrm{s}\\
&\approx 0.7\,\mathrm{Ns}
\end{align}$$
So, the impulse delivered by the human holding the bat, in the best case, is about 5% of total impulse: realistically it will be much less.
This does not show that the human does not affect things like the direction and detailed trajectory of the ball after it is hit: it does show that their contribution to the change in velocity of the ball happens almost entirely before the impact: their job is mostly accelerate the bat and get it into the right place.
It turns out that Dan Russell has a nice summary page, with references on how much the person holding the bat matters. The last two sentences from that page are:
Measurements and computer models show that the collision between bat and ball is over before the bat handle has even begin to vibrate and the ball has left the bat before it even knows the handle exists. Finally, experimental evidence comparing the effect of different grip conditions on resulting batted-ball speed conclusively shows that the manner in which the handle is gripped has no affect on the performance of the bat.
He has a lot of other useful information on the physics of baseball.
Best Answer
In principle yes, but the effect is usually marginal. It also depends on how powerful your lights are compared to the size of the kitchen (a 1000 Watt flood light in a home kitchen will probably have a noticeable effect on the speed of drying).
The question essentially boils down to how much the temperature of the air increases in the kitchen by leaving the lights on. This however not only depends on the power of the lights (one can assume that all power is converted to heat in the end) but also on the size of the kitchen (how much air needs to be heated up) and the thermal isolation of the kitchen (how much heat goes e.g. through the window).