Why did Bree want to wait before going to Narnia?    

From the article, I think Bree wanted his tail to grow again before going to Narnia. Then he can left a good impression to his fellow horses.  So perhaps he was a little vain in his good look.

Why did Aslan give Aravis the scratches on her back?  

Because Aravis drugged her stepmother’s slave to sleep and she escaped, the slave was tortured with stripes laid on her back. So Aslan gave Aravis the scratches to make sure her to know what it felt like. That let her know not to hurt others because of selfishness

Who is Prince Cor?  

It's Shasta!"

How did Prince Cor come to being Shasta?

First, Shasta and Corin were twins. When Shasta was an infant, a cap called the Lord Bar kidnapped Shasta when he heard Shasta was going to save Archenland from a great danger. The Lord Bar was a spy to sell secrete information to Tashbaan. Bard gave Shasta to one knight and sent both away in the ship’s boat. But the knight had starved himself to keep Shasta alive.  Then a fishman adopted Shasta many years. Now Shasta came back to Archenland and saved Archenland from Tabadash’s two hundred horses. So Shasta got back his name of Cor and he became a Prince the same as Corin.