Virgin America only has this weird arrangement of 25% Virgin Group, 75% VAI partners because the US government is protectionist and allows foreign entities to only own 25% of a US airline. When Virgin America started up, there was a long palaver about the CEO, who had to step down after Virgin America started flying, because he was, according to the DoT (and the incumbent airlines), 'under the influence of Virgin Group' (and they argued that it would give Virgin Group more than 25% of influence on Virgin America). Which was complete bollocks, but hey, whatcha gonna do. Other conditions were the dropping of the requirement to have the Virgin logo on the tail (so Virgin America can drop it if they want to, but they won't), and the loss of a board seat.
V Australia is called such so as to not clash with any trademarks and license agreements, since, as someone else pointed out, SQ has the rights to the Virgin name on international routes (through VS). V Oz would be flying from Oz to the US and in the future to South Africa and other Asian destinations, competing with the VA trademark.
Pacific Blue is called such because of the same agreement. SQ vetoed the use of the Virgin name by the airline (started up by Virgin Blue), since it does serve multiple Pacific destinations and is as such an international airline. Virgin Blue is primarily domestic and does not clash with SQ license agreements.

S.