Yes, the deodorant contains a mixture of low boiling point alkanes, such as butane, that form a liquid under pressure but evaporate when the pressure falls to one atmosphere as they leave the can. It's primarily the latent heat of vaporisation that reduces the temperature and makes it feel cold.
In addition to this, deodorants (as opposed to antiperspirants) contain ethanol - they are essentially just a solution of perfume in ethanol plus propellant. The ethanol evaporates on the skin and again the latent heat of vaporisation cools the skin.
In a can of deodorant the dip tube goes down into the liquid propellant. When you press the button the pressure in the can forces liquid alkane up the dip tube and out. The alkane mostly evaporates in the tube and nozzle, but if you hold the can very close to your skin you can get liquid alkane on the skin. This evaporates very rapidly, and it's really cold!
I'm fairly sure that the ridges exist to hold an inventory of liquid. A simple way to argue this is to note that the ridged loop has a much larger total surface area. However, I'm sure there are more complicated physics with the ridged surfaces. Since it helps brace the liquid within a ridge, it's sure to hang on to more water than what a flat surface of equivalent area would.
I presuming that these somehow make the bubble creation process "better" (where "better" may need to be defined)
Indeed, this shouldn't obviously affect the propensity of the loop to fill with a membrane after dipping in water. My poor recollection of playing with these somewhat confirms that. If you dip the thing in water, then at the moment you retrieve it, both loops are likely to have a membrane over them.
Blowing into the loop, producing bigger bubbles, is where the obvious benefit is. In this process, the flat circle transforms into a larger partial sphere (and then a full sphere). That means that the surface tension energy within the free membrane has increased. Comparatively, the liquid within the ridges are at a lower energy state, and as such, it flows into the free membrane. Physically this is justifiable on the basis of natural system's tendency to minimize energy.
So if you want larger bubbles, you need more/deeper ridges. This doesn't have much to do with the propensity of the liquid to "catch", forming the basic membrane within the loop. But as the bubble grows, it sucks liquid from the ridged area. The less liquid you have to create the bubble with, the thinner its walls will be. The thinner its walls are, the more surface tension it has per mass, and thus, the greater the influence of destructive forces will be, and the greater the relative impact of external perturbations.
Best Answer
This behaviour is well explained by Barlow's formula, even though the English Wikipedia article is incomplete in this context. The German version, on the other hand, gives the full picture (which I will quote in the following).
The walls of a pipe (or a similar cylindric container, say, a sausage) experience two types of stresses: Tangential ($\sigma_{\rm{t}}$) and axial ($\sigma_{\rm{a}}$). For given pressure $p$, diameter $d$ and wall-thickness $s$, the individual stresses can approximately be calculated from $$\sigma_{\rm{t}} = \frac { p \cdot d } { 2 \cdot s }$$ and $$\sigma_{\rm{a}} = \frac { p \cdot d } { 4 \cdot s }.$$ Here, you can directly see that the tangential stress will always be larger, which is why it is likely that cracks in the container/sausage will first form in this direction. In fact, this is why the first formula is often stated on its own, just as it is the case in the English Wikipedia article.
Fun fact: The sausage example is used by many German students as a mnemonic helping to remember which of the stresses is larger. As a result, the formulas are often called "Bockwurstformeln" (sausage formulas).
Edit: In response to the comments below, I will try to summarize some details about the above formulas