I would not use steel wool. I've been told that steel wool embeds into the stainless and can get rust started. No idea if its true or not, but, it kind of makes sense and, theres no need to use steel on steel.
Yes, this can happen. First off, Stainless Steel is not really 'stainless'. An alternative, and more descriptive name is Corrosion Resistant Steel (CRES). The Chromium content must be at least 10.5% to classify as Stainless. The Chromium in the alloy forms a passive layer of Chromium Oxide on the surface of the steel. This layer is just another form of corrosion. But as with most oxides, the microscopically thin Chromium Oxide layer forms a barrier to atmospheric oxygen. The oxygen cannot get through the Chromium Oxide layer, so no Iron Oxide, or 'rust' as we know it can form. If the surface of the CRES is broken, either by machining, or even by a scratch, a new layer of Chromium Oxide automatically forms, again barring atmospheric Oxygen from reaching the surface, and again, no rust forms.
If Stainless is cleaned with ordinary steel wool, tiny fragments of the steel wool can break off and be imbedded in the surface of the Stainless. These fragments are not protected by the passive layer of Chromium Oxide and will rust. Then since they are imbedded in the body of the Stainless, the Stainless can start to corrode where it is in contact with the oxidized steel wool.
As far as keeping a Stainless gun clean, yes, the clear shiny surface will show dirt and fouling that might be hidden by the dark blue of carbon steel.
Another myth is that Stainless is somehow easier to clean than carbon steel. A complete myth. Any steel's ability to hold onto dirt is related to surface finish. The more tiny nooks and crannies there are, like with a matte finish, the more stubbornly dirt and fouling will cling to the metal. Given the same level of finish, it will require exactly the same amount of elbow grease to clean Stainless as it will carbon steel.
Lastly, I always say, if you are concerned about the oxide rings that form on the front face of a cylinder, whether it is Stainless or blued carbon steel, you just don't have enough guns yet. Once you have enough guns, you will stop being concerned with carbon rings on the front of the cylinder.