Why can we sniff so much? What was the point for us humans (or other mammals) to develop such a strong sense of smell? Depending on the source, we can distingiush from 10.000 to even trillions of smells. But how was the ability to sniff "artificial" smells (e.g. paint or burning rubber) being useful to our ancestors?

Smell samples the chemical composition of the environment, the groups of molecules you are sampling in those examples still exist in nature. It is like asking why being able to taste pizza would evolve, it doesn't, not by itself anyway, it is the inevitable result of being able to taste the things and conditions that make up pizza, carbohydrates, proteins, fats, oils, acids, bases, salts, etc. things that would be important for our ancestors to be able to detect.

With paint you are smelling volatile organic compounds, and many are important biological and/or hazard markers, some are related to rotting meat, others to infection, others to food sources, and others to natural toxic water sources. With burning rubber you have many different smell depending on the rubber, but in each case you are smelling a burning organic polymer which releases organic compounds which again would exist in nature.