I have noticed that depending on the size of screen I am looking at sometimes the whole slider image is not visible and sometimes it is, is there away of it resizing automatically depending on the screen you are viewing the site from?
Ideally, I would recommend background image size of 2600×1500 pixels, since a sizeable fraction of Internet users have 2560×1440 px monitors these days. JPG is the best format in terms of file size and hence load speed, especially if you optimize and compress it slightly. In general, I would say that a single background image should not be much larger than 500 kB in size.
Regarding your second question – you can find the background resizing settings in “Appearance -> Background Slider”, “Size and position” tab ;)