A slight tilt will not hurt the unit, but a tilt that wasn't there when it was installed can put pressure and stress on the copper lines running to it from inside.
How level the unit is has more to do with how it looks than how it functions as long as it doesn't get too much.
Put whatever under the corner that is sinking, again it's more about availability and what it looks like than anything. Of course you should put something under it that will be permanate.
When moving the unit to do this, remember the copper lines. The are not very flexible and have the ability of putting a lot of pressure on the valves they connect to on the unit.