It can only be placed on fields after you have researched the Mathematicstech. Each sawmill costs 5 stars.
You will still get the +1 population if you build a lumber hut adjacent to the sawmill after the sawmill has been built.
A lumber hut contributes to each sawmill it is adjacent to. For example, a lumber hut built in City A adjacent to City A's sawmill will give +1 population to City A, but if it also happens to be adjacent to the sawmill of City B, then City B will also get +1 population.