Each of these short stories develops more than one conflict. Each has an internal and an external conflict. Internal conflicts are a struggle the protagonist (or other character) experiences within themselves while an external conflict is one the protagonist (or another) experiences between themselves and someone or something in the external environment.
In "Diary of a Madman," the protagonist has an internal conflict within himself as he battles to keep his grip on reality while doing the dehumanizing work of a titular councillor for the arrogant and proud Director of his bureau, His Excellency, with the odd hair that he sometimes combs "upward." He experiences external conflict with the demands placed upon him by His Excellency and by the tedium of the demeaning tasks of his job.
I am a nobleman! I can also work my way up. I am just forty-two — an age when a man’s real career generally begins. Wait a bit, my friend! I too may get to a superior’s rank; .. But I have no money--that is the worst part of it.
He has an additional external conflict related to His Excellency's daughter with whom he has become infatuated and whose movements he likes to follow. These conflicts all converge in the protagonist's decline into a created reality in which he rules as the King of Spain. These conflicts are critical to Gogol's theme about the inherent though abused and disregarded nobility of humankind.
In "Haircut," the protagonist turns out surprisingly to be Paul, the local boy who suffered a blow to his head and as a result has some cognitive impairment. Around him, conflicts rage. In external conflict, Joe pits himself against Julie, who consistently rejects him; in internal conflict, Julie becomes enamored of Doc Stair, who sees her only as a nice young woman; in external conflict, Joe devises a diabolic joke against both Julie and Doc Stair. Paul's ultimate conflict is both internal and external as he first decides how to think about what Joe did to Julie and Doc Stair, and as he then decides what to do about Joe. He resolves the internal conflict by denouncing what Joe did. He resolves the external conflict by punishing Joe. This ironically raises a conflict for Doc Stair who resolves it in favor of Paul.
Paul had came runnin' up to the house a few minutes before and said they'd been an accident. ... Doc examined the body and said they might as well fetch it back to town. They was no use leavin' it there or callin' a jury, as it was a plain case of accidental shootin'.
In "The Lament," the mourning cabby in St. Petersberg has internal conflict as he battles with his grief over his son's death. He also meets external conflict while trying to find a listener who will allow him to give voice to his suffering and to his son's human dignity and honor. He finally resolves both internal and external conflict by telling his story to his cabby horse. His internal conflict is resolved as he has given voice to his lament. His external conflict has been resolved as he has been conquered and defeated in the external conflict by his human passengers. Thus he has turned to his four-footed friend to listen and to give compassion and to, ironically, validate his son's humanity and dignity.
But listen, mate--you know, my son is dead ... Did you hear? This week, in hospital ...
Iona looks to see what effect his words have, but he sees none .... It will soon be a week since his son died, and he has not been able to speak about it properly to anyone.