The name 'stainless' is somewhat misleading: under the right conditions this steel can corrode too. The real mechanism is the passive chromium oxide layer.
The name 'stainless steel' gives the impression that this material will never corrode under any conditions; this is not entirely true. The correct term is 'highly corrosion-resistant'. Under the right conditions this resistance seems almost limitless, but with the wrong environment and the wrong grade choice, stainless steel can also be damaged over time.
The secret of stainless steel is that its chromium content (generally at least 10.5%) reacts with oxygen in the air at the steel's surface to form a thin, invisible chromium oxide layer. This layer isolates the iron underneath from contact with oxygen and moisture.
Even if the surface is scratched or worn, the layer re-forms itself the moment the chromium comes into contact with air again; that is why it is called a 'passive' (self-healing) layer.
The passive layer can fall short under heavy chloride exposure (seawater, salt, some cleaning chemicals), continuous mechanical wear, or wrong grade selection (for example, using 304 in a chloride environment). The resulting pitting corrosion does not mean the material is rusting in general; it is usually the result of a wrong grade/environment match.
As the chromium content rises, and as additional elements such as molybdenum are added, the resistance of the passive layer also increases. This is why molybdenum-bearing grades such as 316 last much longer than 304 in chloride environments. Choosing the right grade code for the right application is as important as the steel itself.
No, not in the literal sense. With the right grade and environment match, corrosion resistance is very high in practice, but in the wrong environment (especially chloride-containing) every stainless steel can be affected over time.
Generally no: the passive chromium oxide layer re-forms itself. However, in combination with continuous wear or contact with aggressive chemicals, the rate at which the layer re-forms may fall short.