there is no such thing as a "fighter's house". there is a "fighter's room", a "fighter's flat", and a "fighter's mansion". regardless of which you have, the "thief's den" i believe has the largest storage capacity, at 250kg.
regardless of if you switch or not, you will eventually find that even 250kg isn't enough space, if you plan to store things. but beware, unless you can buy at least lvl 400-500 traps (if not 600-700), robbers will hit you from time to time. even a lvl 800 has been broken into once. once 250kg isn't enough (or if you'd rather have benefits from a fighter or mage house, and their capacities aren't enough), extra houseparts are the only option. look up the recipe on larkinorguide.com they're relatively expensive to make, for each only adding 1kg. but if you're made of money, and just want a place to store things (apart from the market), it's an option.
and to answer your other question... stuff does transfer from one house to another. even traps transfer, i believe. what i haven't tried is going from a 3-trap house to a 2-trap... i expect you'd lose your 3rd trap. and i also haven't tried going from a full 250kg house to a 170kg (or anything else lower) house... i don't know what you'd end up losing, but i'd bet you'd lose some stuff... who knows what though. (time for sandy or someone else to test, who likes to test things?)
hope these answered helped, and didn't put you to sleep from being too long.
--jason