1 & 3 are correct. Piracy has proved (to be) very hard to stop.
The part after "proved" is the same structure as "this game is fun to play." We could think of this as the short form of "Piracy has proved very hard (for governments) to stop. The verb "prove" does not have to be in the passive. To give a common, short example: "The story proved true." In English, a passive or reflexive meaning is often understood even when not stated explicitly. Eg: The police stopped the thief. The rain stopped (not was stopped).
Btw, Americans usually say "proven" as the past participle, but "proved" is always used in British English.