That was a big in-joke - lucky by name, lucky by nature. ALF ate everything in sight on the show (and most stuff the Tanners tried to hide, lol). But he never ate Lucky (although when Lucky died he did want to digest the remains), ALF ordered Kittens (from an ad, like ordering a Pizza, lol) but when they arrived he found they were so cute he couldn't eat them and he was a CAT LOVER .. much to his disgust!
I remember a couple of great puppetry scenes in ALF - firstly when ALF electrecuted himself in a bathtub trying to use an electric hand blender to make a jacuzzi, (which was only ever shown once and re-shot because NBC complained) and they had ALF come out literally smoking with his hair on end, and there was a scene where ALF cooked dinner for the Tanners and left the gas on, so he ended up blowing up the kitchen and they had ALF fly about 50 feet across the room.