Actually, butlers are supposed to deduct money after every day, or everytime you leave your house and come back to it (at least that's the case for me anyway). If your game isn't even paying the butler even at every day, then there must be a problem. As for the other thing, do you have other households in your game that have butlers too? It's well known that butlers often switch around between multiple households due to the game never having coded a way to mark said butler as your household's sole butler. You may also experience this when you've fired/dismissed previous butlers but let them stay in your game as townies, but of course they would then still keep being butlers and would switch as your butler often because of this.