Can I prevent captions from loading on small devices only (phone size)?
Unfortunately, there are no configuration options available that you could use to do this.
You'd need to maybe use CSS media queries (in your site's 'custom.css' file) or the Juicebox-Pro API (specifically the getScreenMode() method) to detect when Small Screen Mode is being used to display the gallery (in your site's 'custom.js' file) and then hide the caption area, if appropriate to do so, via CSS (although it's not something I've tried so I don't know how successful it would be).
And can captions be loaded differently on various sizes? For instance on the desktop I have them below thumbs, but could they be overlayed on the image just on the phone size?
As long as your Juicebox-Pro gallery sets screenMode to AUTO (and allows Juicebox to determine the most appropriate Screen Mode for the device being used to view the gallery), then Small Screen Mode will be used on mobile devices where the captions will always be displayed on the overlay (on top of the image, where they can be toggled on and off with a tap to the screen).
captionPosition is a Large Screen Mode only configuration option. (It is ignored in Small Screen Mode.) If screenMode is set to LARGE (to force the gallery to be displayed in Large Screen Mode in all browsers and on all devices), then the captionPosition value will be used in all circumstances (across all browsers and devices).
(More information about Juicebox and Screen Modes can be found here.)