The building card stays in play, put it beside your camp somewhere. The materials used in the building are out of the game.
We often stack buildings so just the text half shows and they don't take as much space. Materials are "out of the game" but stay under the building. There isn't a discard pile or anything (although you can of course do it that way if you want).
2) Playing architect/craftsman and using a matching client is the only way to build an out-of-town site.
This is overly strict. The condition is that you need to spend two consecutive build actions on the same turn. Doesn't matter how you got them. For example, Patroning with a Bath will also let you start an out-of-town site if you get two build clients and use them for that. You just need a pair of build actions to bundle.