Hi Simon,
I would strongly recommend design the logo in SVG format, as it can be enlarged or shrunken without loosing quality.
If you are using an image logo with JPG/PNG format, it’s not so great to change its size freely with CSS or customizer.
If so, can the settings in the customizer be left alone?
No, the size from customizer will still be applied to the logo.
Is the size theme specific for things like mobile, tablet and desktop?
It depends on your header/primary navigation settings, it’s kind of complicated, but yes, you should be able to use different sizes.