Why are Sirius Black, Remus Lupin, and Peter Pettigrew named as they are?
Sirius, the brightest star in the sky, was given its name because it is part of the Great Dog constellation; this large star moves with the seasons and was used by the ancient Egyptians to set calendars. The name Black describes Sirius Black's dark hair and at times his dark humor. The name together is what he is as an animagus, a large black dog. Remus Lupin is an entirely wolf-related name. Remus was one of the legendary founders of Rome, who was suckled by a wolf. Lupin comes from lupus, which is Latin for wolf. Peter Pettigrew's name is a bit trickier, but Peter is often a nursery rhyme or fairy tale name for a small, slightly mischievous boy. Peter Pettigrew was certainly small, and he followed his larger friends in their mischief. Pettigrew can be divided into "petti" (petty) and "grew" (past tense of grow), which may suggest that he used pettiness (turning in his friends) in order to feel that he has grown (shared some of Voldemort's power).
Do any signs in the book suggest that Harry might teach himself to become an animagus?
One indication that Harry may try is are that his father did it, and most of the facets of Harry's Hogwarts life parallel things his father did. Harry may follow his father's footsteps in this venture as well. With the help of Hermione, Harry could certainly figure out how to do it-however, Hermione may be reluctant to break that rule, so Harry and Ron may be on their own. There are still worlds around Hogwarts that Harry may want to explore further, such as the lake (which as a human, he can only visit for short periods of time, as he does in the fourth book), parts of the forest, and the skies above the castle; a new form would allow him this.
What is this book's ultimate take on Divination? Does any one character represent this belief?
Throughout the book, Sibyll Trelawney is shown to be quite incorrect in her predictions; in fact, the only times she is right are when she has spotted a pattern and deduced, such as Neville breaking teacups. As we see with Hermione's time-turner, any given event can have a number of different outcomes, and so counting on only one happening becomes increasingly difficult. McGonagall and Hermione are very skeptical of Divination, Trelawney and a few faithful students believe in it, and the centaurs in the Forbidden Forest believe in it, but keep their observations to themselves. Dumbledore seems the closest to summing up its role in the book when he says in Chapter Twenty-two "Hasn't the experience with the time-turner taught you anything, Harry? The consequences of our actions are always so complicated, so diverse, that predicting the future is a very difficult business indeed Professor Trelawney, bless her, is living proof of that." His tone is humorous and kindly toward it as a field of magic, but he doesn't trust it to indicate how life may turn out.