Is David Copperfield portrayed as courageous and sentimental in chapter 13?

This question asks for you to find textual support for two different character traits: courageousness and sentimentality (or tenderness and a propensity for nostalgia). Let's look at each of those separately.

You could argue that David's journey in chapter 13 demonstrates courageousness itself. He walks a great many miles, completely alone, and has little means of providing for himself along the way. He doesn't know what will happen once he reaches Miss Betsey's house, yet he is determined to complete the journey anyway. He also encounters several tough conflicts which demonstrate at least a bit of courage, particularly considering his age. He perseveres in his attempts to sell his jacket even though the shopkeeper violently grabs him by his hair and smells "terribly of rum." When the man refuses to give David his money, David sits outside his shop, courageously facing this drunken "madman" as he awaits a fair payment.

David then encounters a "tramper" who is traveling with a woman with a black eye; readers can presume based on the man's conduct that he has beaten her. When he demands that David give him enough money for a "pint of beer," David watches as the woman silently mouths the word "no" to him. Although the man presents a clear physical danger, David courageously refuses to submit to the man's demands.

There is at least one example of sentimentality. When David hears church bells ringing, it conjures up memories of the past when he had heard bells at Yarmouth:

I never hear the name, or read the name, of Yarmouth, but I am reminded of a certain Sunday morning on the beach, the bells ringing for church, little Em’ly leaning on my shoulder, Ham lazily dropping stones into the water, and the sun, away at sea, just breaking through the heavy mist, and showing us the ships, like their own shadows.

These warm and fond memories are a great example of sentimentality. Notice the feelings of happiness and contentment that are associated with these memories.

I hope this helps as you continue to evaluate this chapter for further support of David's courageous behaviors and sentimental thoughts.

Is David Copperfield portrayed as sentimental in chapter 13?

To be sentimental is to be motivated by emotions, and we can see several instances of David's sentiments in this chapter. David was robbed in chapter 12. In chapter 13, he must make his way to his aunt's home in Dover by pawning clothing for food and sleeping outdoors in rough conditions. He is often hungry and cold. He sustains himself through his emotions, for example, gaining the courage to go on to find his aunt by the sentimental image of his now dead mother being treated kindly by the very aunt he is seeking out:

in her youth and beauty, weeping by the fire, and my aunt relenting to her.

When he gets to his aunt's house and is welcomed into it, he breaks out of his child's persona and becomes the adult author, pen in hand, remembering with fond nostalgia the many items he associates with his aunt's home, such as

the kettle-holder, the two canaries, the old china, the punchbowl full of dried rose-leaves, the tall press guarding all sorts of bottles and pots.

We can him imagine him tearing up at the memories. He also, in his child's self, vehemently defends old Peggoty, showing his great feeling for her.

At the end of the chapter, he shows that his sentiments are alive and that he will not forget how it felt to be one of the unfortunates of the world:

how I prayed that I never might be houseless any more, and never might forget the houseless.

Sentimentality in writing came to be looked down on in the twentieth century, but for writers like Dickens, it was a key component of their work. Dickens, outraged by the conditions of early industrialism, when there was no social safety net, wanted to arouse people's emotions so that they would help those less fortunate than themselves.