Short answers, pulled from the story.
An armorial achievement is the full display of all heraldic components to which a bearer of a coat of arms is entitled. It includes the shield (escutcheon) at its centre and surrounds it with elements such as the helm, crest, torse, mantling, supporters, motto, and any coronet, badge, or order the bearer holds.
In strict heraldic terminology, a coat of arms refers to a garment with the escutcheon or armorial achievement embroidered on it, not to the full armorial display. Using coat of arms to mean the full achievement is common in everyday speech but technically incorrect.
Hatchment derives from earlier English forms such as atcheament and achement, which came from the French achèvement. That noun traces back through the French verb achever, a contraction of à chef venir, and ultimately from the Latin ad caput venire, meaning to come to a head or to accomplish.
Originally, hatchment was used as a general heraldic term identical in meaning to achievement, denoting the full armorial display. King Henry VIII's statute for the Order of the Garter used the spelling hachementis in that non-funerary sense, requiring each knight to display his arms and hachementis on a metal plate at his stall.
Supporters, the figures standing to either side of the shield in an armorial achievement, belong only to bearers who are specifically entitled to them. In modern heraldic usage, baronets are generally not among those entitled to supporters.
In modern English, hatchment refers almost exclusively to a funerary hatchment, a diamond-shaped memorial panel displayed to mark a death. Achievement has taken over for non-funereal heraldic contexts, even though the two words share the same etymological origin and were once used interchangeably.