1. The children were very insistent that they were the help and could help. However, was that the only reason Aslan sent returned them to Narnia after the horn was blown? If not, what other reason(s) is/are there and which reason is the most important?
Going along what Aslan says to Jill in the Silver Chair about them wanting to get to Narnia, I think it's more that Aslan deemed it the right time for the horn to be blown to summon the children into Narnia, but the real, underlying reason is because Aslan willed it. I think the horn was what brought them into Narnia, but it was just the method - Aslan was the one who made the arrangements for the children to be summoned now, and in these circumstances.

2. How did the characters change throughout the book?
Caspian goes from being very naive and innocent to much wiser and braver, though he never loses his hope nor innocence. As we see at the end, when he tells Aslan he's not really sure he's ready to be king, he still hasn't really gained his confidence, at least not yet.
Trumpkin starts off as very skeptical - about everything. Slowly, though, he's swayed to the right side, first by being convinced to help Caspian become king (despite him being a hated Telmarine), then to obey his King's orders despite it running contrary to his own disbelief, the finally believing in the children, and Aslan himself. I think he becomes a more humble character, but also a more hopeful one - I don't think the old Trumpkin would've been able to believe that Peter's "single combat" ploy would have worked.
For the most part, I think the children all grow to be a bit more like their Kingly and Queenly selves, even as children. They're all forced to be mature and grown up again, without the advantage of the years to grow up. They have to take much more responsibility for their actions, while relearning their faith in Aslan.

3. What theme or themes to you see in this book?
Faith in the unseen is a huge theme. Caspian and Cornelius both believe in the Old Narnians they've never seen, Trufflehunter and Trumpkin believe/don't believe in, respectively, the Kings and Queens and the other old stories, and everyone's faith in Aslan is tested before he makes his reveal.

Also, as You Will Find Me Bigger pointed out to me in a conversation we had a while ago, there are a lot of points where, while they need to trust in Aslan to put everything to rights, the characters all have to take action and make their own decisions. The children have to fend for themselves for a while in the ruins of Cair Paravel, rather than having help provided to them right away as it was in LWW. Caspian has to begin the war before Aslan or the Kings and Queens arrive. Even after Aslan has made himself known, Peter suggests the single combat as a delaying tactic, trying to keep the Old Narnians from being crushed before Aslan makes his move. The characters cannot afford to wait for the help to come, though they do depend on it; they have to act and wait.

4. Any other final thoughts?
Of the seven books, I think this one is the least "fantastic" - in the sense that it's related to the word "fantasy". While we still have Talking Beasts, Dwarfs, and the dryads and naiads, they're both not new concepts to those who have read LWW, as well as not really in a large portion of the book. Most of this book is political and faces the realities of fighting in a rebellion, almost up until the end. I just find this interesting because, going through the stories again, it's making me realize that while they're all adventures, they're all very different kinds of adventures from each other.

1. The children were very insistent that they were the help and could help. However, was that the only reason Aslan sent returned them to Narnia after the horn was blown? If not, what other reason(s) is/are there and which reason is the most important?
It occurred to me on this read through just how much of the Pevensie's time was spent trying to get to the Old Narnians. Once they got there, they did very little. Lucy and Susan stayed with Aslan. Edmund helps with the Hag, the werewolf, and Nikabrik. Peter joins in that fight as well and then fights single combat with Miraz. But, it was the trees, which were awaken by Aslan, who did most of the fighting. All of this seems to point to a deeper reason. The children return to Narnia with the confidence in their abilities they would have had as kings and queens, and they still have some of their abilities, like archery and fighting; however, they seem to have forgotten some of the more mental skills they must have learned. They are still relating to each other as children and siblings. Lucy needs to remind Edmund that Peter is the High King. While Peter shows some of the wisdom and leadership he must have have as High King, he did often did not show it. Trumpkin often seemed like he was more of a leader than Peter and showed more wisdom. He thought of moving the boat and pointed out that the river might be the Rush. In one of his lowest moments, he has to be reminded by Trumpkin that he is the High King and needs to vote! Peter is not acting very magnificent; while Susan's gentleness is apparent in her desire not to kill the Telmarines and the bear, she is far from gentle with her siblings; Edmund could probably have been more just in his treatment of Trumpkin; and Lucy is not very valiant since she complains about the task Aslan gives and doesn't have the strength to do it until he breaths on her. In the first book, Peter, Susan, and Lucy are journeying to Aslan, knowing they needed his help. In Prince Caspian, Aslan becomes sort of an after thought. Not to all of them - Lucy says in chapter 8, "But we want to be here don't we? [...] if Aslan wants us." Lucy clearly sees Aslan's paw in this adventure so it is no surprise that she is the first to see Aslan. The children learn on their journey and become more of their royal selves. Peter journeys from a weak leader to the High King in command. It takes some blunders (which he willing admits) and following an Aslan he doesn't see until he gets there. Lewis mentions that Aslan had breathed on Edmund and "a kind of greatness hung around him." I'm assuming that Aslan breathed on Peter as well and that completed the transformation into the High King. Edmund goes from being a passive supporter of Lucy to an active one - agreeing to go with her if no one else will. Susan goes from listening to fears to being a little bit braver after Aslan breathed on her. It is interesting to note that she only feels a little braver - she isn't letting Aslan take all of her fears.

Lucy learns that she can and will likely need to take a stronger stand in what she believes in. It is no longer enough for her to just stick to her story - she must act on what she sees and knows to be true. Unlike Susan who just becomes a little braver after her encounter with Aslan, Lucy becomes a "lioness." Lucy must learn that although Aslan can rush in and save the day, he doesn't always work like that. In LWW, the children were accepted with open arms and no one doubted them. Aslan was believed to be real. In Prince Caspian, they are doubted and undervalued and they have to deal with people who do not believe in Aslan. The children must all learn to increase their faith in Aslan. Lucy learns that her faith must be strong enough for her to follow him even if no one else will. The rest learn that they might not always be able to see Aslan but things will be much better if they follow him anyway. Edmund understands that much quicker than the rest. I feel that the lessons the children learn are more important than the help they give Caspian. Aslan could have chosen other ways to help Caspian but he chose to send the children - not directly to Caspian's door but to a journey where they must rely on themselves (finding food, starting toward Caspian) and Aslan (finally finding Caspian).

2. How did the characters change throughout the book?
See above for the children
Trumpkin's main change is going from disbelief to belief. Caspian starts out very naive but moves along on the journey to understand the world and leading a kingdom.

3. What theme or themes to you see in this book?
(Some of what I think about themes is worked into the responses for #1) Faith, things don't happen the same way twice, Follow Aslan

4. Any other final thoughts?
I think I covered most of my thoughts in answering question 1.