Native screen resolution rather than size is all you need to be concerned about, and your Sharp panel supports maximum Full HD (1920 x 1080) which the NUC will handle fine. HDMI is digital so is an improvement over analogue VGA, for which you'd need a converter in any case, and also a separate sound connection, whereas HDMI carries both video and sound. But as with all things, YMMV and you should try out the different options to see what suits you best.
Edit: on your second question, the NUC can support up to 4096 x 2304 @ 24Hz over HDMI and 3840 x 2160 @ 60 Hz over DisplayPort (or higher resolutions at lower refresh rates). You can also attach multiple displays, and the supported configurations are listed in the Technical Product Specification doc on pages 22-23.