Usually leveling follows the same scheme as SWE and so compensation closely follows. However, research scientists are paid slightly higher within the band for each level given they are typically working on harder tech. So a higher range overall for research scientists, but if you are a really good software engineer you could be within that range too. Also as a PhD new grad you typically start off at the second level at a company.

PhD grads indeed get to skip at least one level at a company (at least at my firm). In terms of mobility, it really depends on where you want to be in your career. Eventually, you'll want to obtain managerial skills and that means less coding.