How do the hauling changes and minecarts change the elements of fortress design?

It's all up to you actually :) (you never thought you'll get an answer like that now, did you?)

The thing that will have to change is the way you handle stockpiles. Slowly you'll realize that you need a specific stockpile for ... e.g. iron furniture with below exceptional quality. Or maybe you already use stuff like that?

Anyway... minecarts. Minecarts can operate even without lines (as of df 0.34.10) you just need two stations - one to load stuff (best keep this one next to one of your larger general stockpiles) and one to unload stuff (this one gets where you want to get the stuff hauled to). When the cart is full a dwarf will come along and haul it to desired location, really slow though (this might be sort of a bug actually, in the end you've got one dwarf carrying enough stuff for 10). And of course you can build tracks for the minecarts (this way they will get pushed by a dwarf). But keep in mind the fact that carts are actually moving quite fast and anything that comes in their way gets trampled and injured or even killed, be it pet or another dwarf. So it might be a good idea to designate special tunnels for tracks (a minecart needs only one track in both directions, unless you build a complex transport-tycoon-ish kind of transport network inside your fortress) or at least designate the tracks as low traffic areas (from the d-o menu).

That being said - if you've built your passageways to be at least 3 blocks wide you don't need to change your core design ideas. Otherwise if you insist on using carts on tracks you might want to try to expand the passageways and/or try to find alternative routes for designated track tunnels.