The typical guidance on storing enzymes is to keep them in a -20C freezer, but would storing aliquots of enzyme for long-term storage in a -80C freezer cause damage that would result in the enzyme losing functionality?
I always store enzymes at -80oC. It is not even necessary to include glycerol, although if you include 10 or 20% glycerol the frozen solution will thaw faster. I consider -80 to be preferable to -20 for long-term storage.
I divide the enzyme stock solution into small aliquots (e.g. 100 µl) and flash-freeze them in powdered dry ice or a dry ice-ethanol bath or liquid nitrogen. The purpose of flash freezing is to prevent separation of the solutes from the ice.
In almost every case, the enzyme solution can be frozen and thawed many times without loss of activity if it is handled in this manner. Moreover, they can be stored at -80 for many years without losing activity.
That depends on the enzym and the kind of preparation: Enzymes in ammonium sulphate should never be frozen at all, store them at 4 °C. Enzymes sensitive to freezing can be stored in water/glycerol mixtures that stay liquid down to -40 °C or so, these are stored at -20 °C, not at -80 °C. If enzymes in aqueous buffers are frozen, then it is usually best to store them as cold as possible. Do NOT use self-defrosting fridges, these cycle the temperature which will kill most enzymes quickly!