Yes. The second sentence entitles the data subject to have incomplete personal data completed, including by means of providing a supplementary statement, though the measure of completeness is the purposes of the processing rather than an abstract standard of exhaustiveness.

The article itself requires rectification without undue delay, and the general framework of Article 12(3) requires the controller to inform the data subject of the action taken within one month of the request, extendable by two further months where the complexity and number of requests make that necessary.

Yes, within limits. Article 19 obliges the controller to communicate any rectification to each recipient to whom the data have been disclosed, unless this proves impossible or involves disproportionate effort, and to inform the data subject about those recipients if he or she requests it.

Point (a) of Article 18(1) allows the data subject to obtain restriction of processing for the period the controller needs to verify the accuracy of the contested data. If the controller refuses to rectify, Article 12(4) requires it to state its reasons and point to the possibility of lodging a complaint with a supervisory authority under Article 77 and seeking a judicial remedy under Article 79.