According to the Flag Institute:
From its earliest days, the Admiralty often referred to the flag – however it was used – as the Union Jack. In 1902 an Admiralty Circular announced that either name could be used officially. And in 1908 the UK Parliament approved this verdict, stating that 'the Union Jack should be regarded as the National flag'.
Yes, as I understand it the jackstaff got its name because the Union Jack flew from it.
Charles II said all navy ships had to fly a small (or jack) version of the flag.