No, the Bounty films are about Tahiti. The famous mutiny occurred in Tongan waters as Captain Bligh and his crew were on their way back to England. Fletcher Christian, eight fellow mutineers, and 18 Polynesians landed on Pitcairn Island several months later, after an unsuccessful attempt to colonize Tubuai Island south of Tahiti. To avoid detection, they burned their ship in Bounty Bay at Pitcairn after unloading everything of value. In 1791 - two years after the mutiny - the HMS Pandora arrived at Tahiti in search of the mutineers, but it was not until 1808 when an American seal hunting ship chanced upon Pitcairn Island that the end of the story became known to the world.
Most of the fewer than 50 permanent inhabitants of Pitcairn Island are direct descendants of the mutineers and their Polynesian wives. Several attempts have been made to relocate the Pitcairners. During a drought in 1831, they were evacuated to Tahiti, but all returned to Pitcairn within a year. In 1856, all 193 islanders were taken to the former penal colony at Norfolk Island near Australia, which seemed easier to administer and supply (a baby was born on the voyage, so 194 Pitcairners actually arrived). Two families returned to Pitcairn in 1858, followed by four more in 1864. The present island population (excluding temporary residents) is descended from those four families. There are also large numbers of Pitcairn people on Norfolk Island and in New Zealand and Australia.
Pitcairn Islands is the only remaining British possession in the Pacific. It's "islands" because the colony includes uninhabited Oeno, Henderson, and Ducie, as well as Pitcairn. In practice, the islands are owned by the Pitcairners on Pitcairn, who govern themselves by means of a 10-member Island Council. The governor is the British High Commissioner in Wellington, New Zealand, while the actual administration is controlled by the Commissioner for Pitcairn Islands in Auckland.
None. There have been a number of proposals to build an airstrip on Pitcairn but none has yet materialized. An airstrip would destroy a good part of the island's agricultural land and alter the way of life on the island. Yet at last report, proposals for an airport and tourism development were back on the table. Meanwhile the only visitors are people off a handful of passing yachts and cruise ships which can only stop here for a few hours as there's no safe anchorage. A Sausalito, California company, Ocean Voyages, Inc., organizes yacht charters from French Polynesia to Pitcairn Island.
You'll need a residence permit issued by the Pitcairn Island Administration in Auckland, and your application must be approved by both the Island Council and the governor. You'll be asked to provide a specific reason for going, and journalists are subject to special scrutiny. A few years back a British woman named Dea Birkett got permission to spend a few months on the island, and ended up writing a book about her experiences titled Serpent in Paradise. Dea provides a fascinating (and perhaps unrepresentative) glimpse of Pitcairn today, and the authorities aren't interested in a sequel. (Those going on the Ocean Voyages tours will have these formalties taken care of for them.) Of course, if an airstrip is built, Pitcairn's status as one of the world's most remote islands will come to an end.