Dialect is used in Stuttgart, yes :-) I recently talked with a woman from Stuttgart and I was impressed, how strong her local dialect was noticeable for me. At least for me as a northern German it was clearly hearable.

Yes, definitely.

One of the mottos of Baden-Württemberg (the state containing Stuttgart) is Wir können alles, außer Hochdeutsch "We can do everything but except speak standard German". In reference to the various kinds of industry and the variety of exports and to the local dialect.

Stuttgart itself may not be as bad, being a large city and like most large cities, a bit of a melting-pot for people from various dialect areas, but you will hear a lot of local dialect or at least more or less strongly accented standard German in the south.