Cancer is when cells overgrow due to mutations in genes that affect the cell cycle, so cells start to reproduce and replicate without the controls. Control mechanisms within the cell cycle usually detect and repair bad DNA, and they make sure the cell doesn't reproduce at too high a rate, and they make sure that cells die at the right time.

Sometimes people are born with an inherited mutation that may contribute to cancer, but about 90% of cancers are caused by mutations that arise in the individual during their life. There's nothing from looking at a mutation to say what caused it, unless they look at your family members and decide that it was an inherited mutation.

The way they decide some chemicals are carcinogenic is that exposure to them correlates with higher rates of some kind of cancer. But they can't look at the mutation itself and say what caused it.

If you get a type of cancer that they have already noticed is highly correlated with exposure to a chemical you work with, then they might be able to say that your mutation was probably caused by that chemical.

The hydrocarbons (oils, etc...) that mechanics get exposed to are absorbed through skin contact and inhalation. It's found in lymph nodes and can build up over time.

It's generally accepted that auto mechanics have increased incidences of cancer and heart disease. For cancer, increases were seen for pancreatic cancer, urinary cancer outside the bladder, and lung cancers called pleural mesothelioma.

It very much depends on the type of cancer! For an easy example: if you develop mesothelioma and have a solid exposure history to asbestos, they will attribute it to that. For cervical cancer, we are pretty good at telling if it’s HPV-related or not. If you get an odd cancer and have no other risk factors other than your exposure history, it will be listed as a possible or probable contributor in the notes provably.

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Also note that far more things "are known to cause cancer in California" than anywhere else!
For various interesting/historic reasons, California has exceptionally tight environmental restrictions (for pollutants etc), and one of the lowest thresholds for labeling products as "suspected" or "known" to cause cancer.
I'm in no position to judge whether California is "reasonable" or "over cautious", but for people who want to stir up panic, they'll always cite contamination levels (e.g. in water) compared to California's limits!

They would all be probable causes. Carcinogenicity usually just means that certain substances cause cells to undergo mutations, which could then possibly lead to cancer. However some substances are known to cause specific types of cancer due to the nature of how and where we are exposed to them i.e. skin, lungs, systematic, etc.. Certain chemicals also accumulate in different tissues and would be more likely to cause cancer there. Others get metabolized by the liver and only turn carcinogenic then and there.