Corruption is plainly made a focal point in the opening paragraph: "he plainly demonstrated that all the imperial institutions were going to the dogs." The expression "going to the dogs" is an idiom, or an English translation of a similar Russian idiom, that figuratively indicates that something is disintegrating in integrity and quality; it is becoming corrupt. This idea is reemphasized at the end of the same sentence when the third person narrator intimates a justice of the peace was habitually appearing in public "in a drunken condition." In Gogol's day, this scandal could remove a justice from public office. This is a sure indication of corruption.
Futility, an equally important theme in "The Overcoat," is also delineated in early paragraphs, first, in the manner of his being named ("it was a case of necessity, ... utterly impossible to give him any other name") and, second, in his status at the office as titular councillor:
However much the directors and chiefs of all kinds were changed, he was always to be seen in the same place, the same attitude,
the same occupation;
The narrator makes a rather large point of the futility of Akakiy's occupation being seemingly eternal--always doing the same thing, in the same place, in the same way--with the legend having grown up of his having been born there with a "bald head." Futility, of course, is defined as uselessness, ineffectiveness, purposelessness (Collins Dictionary).
Discussing madness in Gogol's story is a little less successful. In fact, if it is asserted that either Akakiy or the "prominent personage" is mad, Gogol's whole point disintegrates. His point is, of course, the inhumanity with which persons in lower stations are treated by persons in higher stations. This was an important theme to Gogol and appears in a number of his works including Dead Souls.
Nonetheless, there may be suggestions of rapid descent into madness in the brief description of Akakiy's fever delirium when he sees visions of robbers under his coverlet, visions of his old mantle (cloak), and visions of the "prominent person." There may also be an earlier suggestion of madness in Akakiy's reaction to the scolding brutally delivered to him by the "prominent person." Akakiy is in such shock and terror that he nearly crumbles to the floor, leaves the building in a daze, and walks with mouth agape through a blistering St. Petersburg snowstorm, which leads to his death.
He went staggering on through the snow-storm, which was blowing in the streets, with his mouth wide open; ...