I asked the same questions to my students when studying this book, and they decided that Sydney Carton was definitely a developing character in this novel. Of course, in a sense, he is the hero or one of the main protagonists of this great work of literature. Key to realise and focus on is how he is transformed by his love for Lucy, even though he knows that his love is hopeless.
Let us focus on how Sydney Carton is presented at first. As the "jackal" to Stryver's "lion," he is shown to do all the hard work without receiving any of the credit. Although he is a man of talent, it is clear he chooses not to apply himself as he needs to to get on in life. Note how he is described at the end of Chapter 5:
Sadly, sadly, the sun rose; it rose upon no sadder sight than the man of good abilities and good emotions, incapable of their directed exercise, incapable of his own help and his own happiness, sensible of the blight on him, and resigning himself to let it eat him away.
This clearly presents a very sad picture of Sydney Carton. Likewise at the end of Chapter 4, Sydney Carton berates himself in the mirror after his talk with Darnay, saying, "A good reason for taking to a man, that he shows you what you have fallen away from, and what you might have been!" Therefore Sydney Carton seems to be well aware of the kind of man he could have been, but for whatever reason, is unable to be.
However, when Sydney reveals his heart to Lucie, in the chapter ironically entitled, "The Fellow of No Delicacy," he shows how Lucie's presence in his life has given him hope, but that he has rejected that hope as a "dream":
"I wish you to know that you have been the last dream of my soul. In my degradation I have not been so degraded but that the sight of you with your father, and of this home made such a home by you, has stirred old shadows that I thought had died out of me. Since I knew you, I have been troubled by a remorse that I thought would never reproach me again, and have heard whispers from old voices impelling me upward, that I thought were silent for ever. I have had unformed ideas of striving afresh, beginning anew, shaking off sloth and sensuality, and fighting out the abandoned fight. A dream, all a dream, that ends in nothing, and leaves the sleeper where he lay down, but I wish you to know that you inspired it."
Clearly the role of Lucie in Sydney's life is incredibly important in provoking that change that he seems to find so unlikely and impossible. And change he does, for, as we know, this man that describes himself as being full of "sloth and sensuality" goads himself into action to bring happiness to Lucie by sacrificing himself. Ironically, in his death, Sydney Carton finally finds the peace and wholeness that eluded him in life. Note how the bystanders described him as having the "peacefullest man's face ever beheld there." Others described him as "sublime" and "prophetic." The unforgettable and moving ending of this novel shows how one man is able to resurrect himself (a key theme in this novel) through sacrificing himself to death, showing that Sydney Carton is undoubtedly a dynamic character that develops as the novel advances.
He is a developing character because in the beginning of the novel he is known as someone who does not care in this world but at the end of the novel Dickens portrays him as a compassionate individual because of his sacrifice to die in place of Darnay.