A typical screen-door spring is tension-stressed in its normal state; that is, adjacent turns cling to each other and resist separation. Does such a spring obey Hooke’s law?

Yes, such spring obey Hooke’s law.
The motion of a body is said to be periodic if it passes through similar conditions after equal intervals of time. Simple harmonic motion is periodic in nature. In accordance to Hooke’s law, with in elastic limits, tension is proportional to extension from the equilibrium position and always points toward the equilibrium position. The spring tends to die out the separation between coils. So it tends to regain its unscratched position. Therefore it obeys Hooks law.