Given how confused the OP is regarding the fitment of the airbag, and how dangerous an airbag is if incorrectly wired, I would strongly suggest OP seeks advice from someone experienced in doing the swap rather than going down the DIY route if they are unsure of what they are doing.
Regarding the other posts:
As a general rule of thumb, Audi flat bottom steering wheels pretty much fit any of their models. Obviously if your moving from a non-multifunction wheel to one that is equipped with buttons, unless you do the extra work, the buttons will not work. Moving from a regular multifunction wheel to a flat bottomed MF wheel, should not be an issue - the only potential hitch is that if you have a previously heated round wheel, the flat bottom is non-heated therefore there is a change in function for the * button. And as an FYI, there are NO heated options for any flat bottom wheel on any model.
Airbags are more complicated. Here is my general experience.
First off, mechanically, airbags are not interchangeable between a round steering wheel and a FB steering wheel. Audi used different fixing methods - despite the fact that both airbags are round in shape.
Secondly, there are different types of airbags as mentioned in posts above: Single stage and two stage (which refers to the firing sequence of the actual airbag mechanism) If you don't fit the right type your going to compromise the safety of the car, probably throw a fault code, and likely fail your MOT. Unless you choose to frigg it, in which case your compromising your safety anyway.
Thirdly, the connectors from the airbag to the steering wheel are different between different models (this is from experience - Ive changed my wheels over on most of the cars Ive had, including the S7) The connecting pins may have similar pin-out functions - but the connector shells are keyed differently - This may not be applicable on a 8P A3 swap to an 8V A3 as Ive not swapped a wheel over on the A3 model.
Buying airbags from eBay is a minefield without part numbers and pre-checking it will fit you car - so be warned. Most sellers haven't a clue what models the bags will fit, and list all models as a default.
Buying a FBMFSW & Airbag from one family model and fitting it to the same family model does not usually present any problems (i.e. C7/C7.5 A6/A7/S6/S7/RS6/RS7)
Swapping between models (i.e. A3 to A4 or A5 etc.) the wheel will fit, the airbag wont (in the majority of cases) You need to confirm the correct airbag part number for your car.
