Can I build a settlement if I don't have the building piece for it?
This is super clear in the rules.

Under Build in the Game Rules section: 


  You cannot build more pieces than what is available in your
  pool—a maximum of 5 settlements, 4 cities, and 15 roads.


Then, later, under Build (Building) in the Almanac:


  You can build as many items and buy as many cards as
  you desire—as long as you have enough resources to “pay”
  for them and they are still available in the supply.
  [...]
  Each player has a supply of 15 roads, 5 settlements, and 4
  cities.


(Emphasis added.)

and again later, under Settlements:


  Note: If you have built all 5 of your settlements, you must
  upgrade 1 of your settlements to a city before you can build
  another settlement. You will then have the settlement in your
  supply, so you can build another settlement.