Do we ever earn new hammers with which to build structures in expansion? Or is the only way to get them to restart the area?
Expansion is currently end-game-ish content. While you can on rare occasions, gain hammers through different Outerworld options, the main way to get them is by winning in the Path of Chaos, upon victory you have a high chance of encountering hammers.
If you haven't seen this yet, it means you haven't traversed far enough into the Outerworld yet.
And if I do start a new area, will I keep my current totals of each of these resources or will everything be returned to its default amount?
Should reset to the default amount, else you could keep re-rolling until you get a more favorable map (my thought process anyway).
I have been playing expansion for less than 24 hours, I think, so I am hoping that hammers are earned each day.
It's one of the more long-term methods of progression in the game, so it will take some time to see progression there.