1. The change of leadership triggers a change of PM without a change of government. As incumbent PM but ousted leader, Gillard is obliged to advise the GG of her resignation and the identity of her successor (as reached in accordance with the internal political processes of the Labor party) (cl 6.51).
2. The GG is obliged to accept the advice, as the government continues to have the confidence of the House. The government continues to command confidence unless and until there is a lost vote of confidence (or some other similar public, clear and incontrovertible evidence). It's not the job of the GG to dip into the political morass to query the various speculative statements about support for Rudd etc. It's the obligation of the political parties to clearly manifest the position - publicly and unequivocally. (cl 2.2 and ch 6.36-6.40)
3. If Rudd subsequently loses the vote of confidence, he becomes a caretaker PM. As a consequence, he loses his mandate to request the dissolution of Parliament for an "early" election. Any decision about the dissolution of Parliament for an early election, including effectively the date of such an election, must be made in accordance with the caretaker convention. As a significant decision, the GG is only entitled and obliged to act on a request if it is made with the support of a majority within Parliament (cl 6.20 and 6.58). The reference to early election means dissolution of Parliament before the constitutionally mandated date - the GG would, of course, be obliged to dissolve Parliament for elections if that date was reached.
4. If Rudd survives the vote of confidence, he is entitled, as usual, to request a dissolution for an election as and when he wishes (cl 6.56).