"Do they lose their leaves?" Yes, over time as the new fronds (leaves) emerge, the older lowest fronds expire, but are always in a constant state of having leaves / fronds. "Do they fall off" Palms hold their dead leaves for varing lengths of time depending on genus and species.
Just to elaborate on what Barrie said, some of them are self cleaners, there fronds fall off by themselves over time, and others don't. Theones that don't form a 'skirt' but can be cut off manually Ed
