There's no problem about the suffix -ery, a variant of -eria as sometimes revived today in such elegant coinages as washeteria. Thus a pottery is a place for making pots, and I suppose the aforesaid mansions are fat catteries, though a buttery is not the place for making what the cats are as fat as, a butt being a receptacle for wine.
The two lots - the first meaning the part, the second the whole - can be heard at the same time in the expression "You've had your lot", in other words you've had the whole of your share and there's no more coming to you, but there's also an echo here of lot in the slightly old-fashioned sense of fate, which is a matter of chance, decided by whatever gods are in charge of such things. The ancient Greeks sometimes made Chance herself into a goddess, with unfathomable motives and a randomly distributed smile. There are those today who near-worship the lottery but certainly not the acolytes who tend it (whom they would probably describe as a greedy lot).