The Case of the Golden Idol/The Dramatic Departure of an Outsider

The screaming man on the left
He is carrying a book with the added note "Property of the Pear brothers", so that is clearly his last name. As for his first name, he is carrying a knife with the initial A. The duty roster on the barn door clearly reveals the names Adam and James. Matching the initial with the roster makes his name as Adam Pear.

The burning man
He is carrying a horse brush and a knife. The brush suggests that he is taking care of the horses. The duty roster on the barn door clearly reveals the names Adam and James. The knife he is carrying having a J engraved on it. With the carving on the wall inside mentioning the "Pear twins" and Adam already identified, this is clearly James Pear.

The shouting gray haired man
The man is carrying a horse betting ticket. Another betting ticket can be found in the fireplace. Adding up the bets, he spent 85 pounds on gambling. Since he doesn't have any pocket change, it is not unreasonable to suspect that he has a gambling addiction and is likely in debt. A letter in the fireplace addressed to a Peter Battley makes it unambiguous that there is a big debt.

One of the scraps from the will speak of financial troubles and a dead mom. The family tree on the attorney gives one gives one son of a dead woman, Peter Battley. This is that man.

The calm black haired man
The will in the fireplace states who the inheritor of the golden idol is. That part of the will matches to the remaining name on the list of inheritors, Willard Wright. Since this man is holding the idol, he is Willard Wright.

The man with a saddle bag
The note on his person suggests that he has been asked to come by an "E.C." to provide a carriage. A letter on Edmund recommends the hiring of a coachman named David Gorran. Going by the fact that this man is coming from the stable and is carrying a saddle bag, it seems like Edmund indeed hired David Gorran.

The man with glasses and brown hair
He is carrying a letter addressed to Edmund. He is also carrying a newspaper talking about an Edmund Cloudsley. He is also carrying an handkerchief embroidered with the initials E.C. This is clearly Edmund Cloudsley.

The woman with a fan
Given that there is only one woman in the scenario her name is by default the only woman name available. The family tree on the attorney lists her name as Rose Cubert.

The man with glasses and gray hair
Going by his dialog he seems to be experienced with legal matters. He seems to have placed his card on the table in front of him, identifying him as the attorney Nicolas Maker.

The first scrap
It speaks of "the colonies", and the woman is carrying a book, a fan and a katar blade, all suggesting visiting foreign territories. This part is addressed to Rose Cubert.

The second scrap
This scrap speaks of politics. Given that according to the newspaper, Lord Edmund spoke in parliament, he is likely involved in politics. This part is addressed to Edmund Cloudsley.

The third scrap
This scrap speaks of financial troubles and a dead mom. The letter in the fireplace addressed to Peter Battley mentions a debt of 255 pounds, a big amount. Further more, the family tree on the attorney shows one man with a dead mom. This part is addressed to Peter Battley.

The fourth scrap
This scrap simply states that who will receive the idol. Willard Wright is the only remaining inheritor, so that has to be him.

Filling out the scroll
One inheritor in particular seems to have gotten shafted, only getting a book of aphorisms, namely Peter Battley. Adding in his debt it is likely that he'd be upset by this matter and take things into his own hands.

Thusly, he ordered the Pear brothers to take the idol for his own benefit. However, the new owner of the idol, Willard Wright knew of what it could do and defended himself. He used the power of the idol to cause James Pear to spontaneously combust.