Here's my take - and I'm not a pro.
During long runs (15 miles plus) on a gradual hill of more than a quarter mile it can help the muscles around the top of your quads and hips area if you do extend the stride for a bit as it stops them (muscles) shortening and becoming tight through repetition. So, not always a case of shorten your stride.
On short steep hill reps or fast runs, keep your foot alignment the same as on the flat - avoid splaying out side ways like a penguin. This will increase your drive and knee lift. Don't worry about the arm pumping, this will follow suit depending on how hard you have decided to run.
On long gradual hill reps and fast runs, keep the foot alignment true and try and keep to your natural cadence as per flat surfaces.
Increasing your cadence on a hill can actually be a good thing depending on the length of the race or the type of terrain. Though, as others have said, some fell runners do walk and I certainly walked big hills in a recent trail marathon in order to save energy.
Down....
First off, a few dip reps (standing on a stair facing down - on one leg - dip so that the free leg just touches the floor at the base of the step and slowly back up * 10 each leg) will help build the muscles needed to let you take the breaks off going down hill.
Speed down hill is - IMHO - safer as your foot is never planted long enough to worry about a slip. Down hill is about commitment in my book.