I've never seen a real GMAT question where items 1 or 2 above might be at all helpful. I have seen one or two test prep company questions where you can use those facts, but it's not the type of thing you'd be expected to know on the real test.

Item 3 is absolutely fundamental in Number Theory. It's an extremely useful fact to understand, and is tested in various ways all the time, at all levels of the test. Item 4 is essentially a corollary of 3, and is also frequently tested (the infamous 'h(n)' question from GMATPrep is just one example). Item 5 doesn't really say anything, but you'd certainly want to know that, when you add two non-multiples of 8, for example, that you can sometimes get a multiple of 8 (5 + 3 = 8).