They do.
As an annecdote, the professor who formerly held the "Machine Learning" course at TU Munich left for VW's AI research group (https://argmax.ai) a few years ago. Whether that qualifies as "science" may be debatable, but the larger players in the automotive space are definitely invested in this kind of research.
From the article: His group’s paper runs down a wish list of quantum chemistry simulations that sufficiently robust quantum computation should be able to tackle. Such problems include designing next-generation batteries, optimizing solar cells via detailed study of photosynthesis, and faithfully simulating complex molecules without resorting to approximations that conventional computers must rely on to make the simulations tractable.
Quantum computing is also of interest to self-driving cars. I know that at least some other big car companies do quantum computing work for that reason. I wouldn't be surprised if this is a test of sorts.
They do. These days, battery chemistry is probably very interesting to them. Earlier, it would have been problems such as optimizing combustion chamber shape, etc.