Dr. Huxtable is a professor and runs a fancy private school for boys. He makes a very dramatic entrance at Baker Street by fainting. Turns out he's worried about a missing student and, hoping to avoid a lawsuit, hires Sherlock Holmes to find the kid. Thorneycroft also gives us some insight into class structures in the 1890s. He is hesitant to act without the Duke's approval, and he's clearly subordinate to aristocrats like the Duke, even though he's a respected professor and head of an important school. In terms of class, Dr. Huxtable would be more middle than upper class.
The Duke is the father of the missing boy in this story, and he acts like a pretty uninterested parent for the most part. In trying to protect his illegitimate son, who committed the kidnapping, he ends up ignoring and mistreating his younger son. The Duke is also more concerned with minimizing any "scandal" than he is with the thoughts and feelings of his sons. Holmes has very limited patience for the Duke and his bad behavior; Holmes even takes his reward money at the end. As Watson tells us, this is a very unusual move for Holmes, who usually doesn't care much about money.
For the bulk of this story we think, along with Holmes and Watson, that Wilder is a private secretary to the Duke. However, as the Duke dramatically reveals later, Wilder is actually his illegitimate son. This is why Wilder decides to kidnap his younger half-brother; he is hoping to blackmail his father into making him the heir instead of the boy. Wilder is a jealous and embittered young man and it leads him to get involved with some shady people when he orchestrates his younger brother's kidnapping. The Duke explains that Wilder has been emotionally blackmailing him for years. Holmes convinces the Duke to throw Wilder out of the house at the end.
We keep wanting to call him Lord Saltine, like the cracker. That is a really unfortunate title, but his real name, Arthur, is normal enough at least. Poor ten-year-old Arthur has a rough time in this story. His mother has moved out, his father, the Duke, is cold and distant, and his older half-brother tricks him and kidnaps him. Then his dad doesn't come get him for a few days since he's trying to protect the older brother/kidnapper. Holmes stands up for Arthur's interests to the Duke, which reveals Holmes's humanitarian streak. Watson implies that Arthur will have a much nicer home life when the story ends.
The Duke's wife is absent for the entire story, but her lack of presence is quite important to the plot. After all, James uses Arthur's mother as bait to lure him out of school. During the Duke's big confession scene, we learn that the Duchess left for France because her husband was so distant and because his secretary/secret son James was driving her nuts.
You may be wondering why she would just up and leave her son Arthur like that. Well, in this time period it wasn't very easy to get a divorce. People did separate though. However, women didn't have legal rights over their children in this era. Children always "belonged" to their father, so unless he agreed, a woman couldn't just take her kids and leave her husband. Presumably, the Duke didn't want Arthur to leave, so he didn't. It's sad because Arthur clearly preferred his mother, and his dad shipped him off to a boarding school after she left. Holmes plays matchmaker in this story as well as detective, and he convinces the Duke to contact his wife and try to reconcile now.
Heidegger bravely chases after Lord Saltire when he realizes there's something fishy happening. He chases after the boy on his bike and is killed by the kidnappers in his effort to rescue the boy. It's by tracking down Heidegger's body that Holmes gets on the right trail and eventually locates the missing boy.
Reuben Hayes is the true villain of this story. He is the kidnapper that James hires to take Lord Saltire, and he is also the man who kills poor professor Heidegger. Hayes is abrupt and rude to Holmes and Watson, and he seems to care about nothing but money and saving his own neck. He tries to bail on James, but Holmes makes sure that he is caught in the end.
Reuben Hayes's long-suffering wife is the one who cares for Arthur while he is being held at an inn during his kidnapping. The Duke describes her as a kind woman, and he uses her in an effort to justify his decision to leave his son with his kidnappers while he sorts out things with James.