No, it is not. It is a transitive verb (to begin, to start up) or a noun (a person beginning instruction or membership). The participles of the verb, initiated and initiating, could both be adjectives.

Home rule cities.

The answer to this question is state specific as it depends on the statutes or Constitution of the state in which the initiative or referendum petition would be filed. My experience is in the Commonwealth of Massachusetts where I served in the House of Representatives. In Massachusetts, the Initiative and Referendum petitions are filed with the Secretary of the Commonwealth. If the person who filed this inquiry indicated the state in which he/she would be filing, we can provide an more accurate response.

Both True and False, Taking Initiative is to initiate something into it being done. Though from the person taking the initiatives point of view this is probably something that needs to be done.

He took the initiative, a peace initiative.