AIRail is a collaboration between Deutsche Bahn and Lufthansa to transport passengers by train instead by aircraft from one airport to the other.
This trips are like flying, since you get a boarding pass including seat assignment for a certain train.
Even the train stations are treated as "airports", since for example Cologne main station has its own IATA code QKL (while Cologne Airport has CGN)
Rail&Fly is an offer of several airlines to travel between any Deutsche Bahn train station (and some foreign ones) and your airport "for free". You can take almost any train including ICE at the day of departure and the day before as well as at the day of arrival and the day after. You can have stopovers, but no major detours.
And you will need to have a ticket, which you can print out yourself (www.accesrail.com/checkin) or obtain from a ticket machine.
That's the most important information in short, for more, see the Rail&Fly website of Deutsche Bahn (English version).
From the information above, it should be clear: In case of AIRail, you are supposed to take a certain train, and your luggage will be transported to the AIRail terminal. But the airport does not have anything to do with Rail&Fly, it even doesn't know your personal schedule. May be, you stay in Frankfurt for a day? Here, you have to collect your luggage at the standard place.
S-Bahn in Stuttgard
Well, the german transport associations... When you are traveling within the area of a single t.a., you have to buy a ticket for that t.a., and can use buses, trams, local and regional trains (but not long distance like IC/EC/ICE).
If you are traveling through more than one t.a. area by train, you buy a DB train ticket, which is only valid for the trains, but not buses and trams.
Of course, if you have Rail&Fly and just have to travel within a t.a., you don't need a t.a. ticket for the trains...
Since the S2 and S3 are trains in this context, you can use the Rail&Fly ticket for them.
There's a trick to find out where your Deutsche Bahn ticket is valid: The Deutsche Bahn website is really good in finding connections even between any two bus stations in Germany, but you can only buy train tickets there. So if the connection states "fares not available" or "partial fare", this means you need a t.a. ticket. Clicking on "partial fare" also shows for which legs.
But as I wrote: If start and destination are within the same t.a., you can't buy a ticket either, so choose one of them far far away to check if a train ticket is valid.