1 7/8" is the standard barrel length for J-frame snubbies; what most folks refer to as a "Two Inch J-frame" is actually a 1 7/8" J-frame.
Of course, Tamara is basically right, but it isn't all the info. The standard barrel on a .38spl S&W small-framed snub (J-frame) is 1 7/8". On the .357s they are 2" or 2 1/4" (it has been long enough since I've looked it up I don't remember which- Tamara probably knows off hand though, she's good with that stuff). Taurus (not technically J-frames since that is a former S&W designation, but for all practical purposes when we are talking J-frames most people include/think all small-framed revolvers) uses a real 2" barrel on their .38spls and a 2 1/4" on the .357mag. I don't know if Colt uses a true 2" or not on their 2" small-framed snubs. Of course, for ease, as Tamara said, for all of these most of us just refer to them as a 2" gun. Then again, since you are asking about the 2" (1 7/8")
.38spl S&W Airweights all but the info that Tamara gave is pretty irrelevant, but what the heck.
Oh, as for the original questions, they have mostly been answered by others, but...:
The Airweights are aluminum alloy. The barrels, and I believe the cylinder, are steel.
The newer ones are rated for +P, the originals weren't. I think you can shoot the newer ones pretty much exclusively with +P and they will last for quite a long time so I wouldn't worry about it. I'd suggest regular .38spl for practice but that is to save your hand not the gun. Now the older guns are a different animal, they aren't +P rated, and on those I'd limit my +P use.
With a shorter barrel there are concerns about expansion. However, there are several older loads that are still popular that are good in a snub (158gr SWCHP, 158gr SWC, and 158gr LHP) are there are also several newer hollowpoints that are reputed to do just fine in a small snub. The new 130gr Gold Dot looks especially interesting.
I don't think a DAO gun will have any worse of a trigger pull than the DA of a SA/DA revolver. Of course you don't have the SA pull as an option if you don't like the DA pull, but since these are meant as defensive guns I don't see that as a problem. Using the SA pull on a defensive revolver will take extra time that you shouldn't be using and if you will be using the DA in a defensive situation you should be practicing with it (this isn't a target gun). Anyway, the DA trigger on my DAO Taurus 85CH is quite nice, so if anything, I'd believe that the DAO revolver probably has a better DA pull than the SA/DA gun does.