The story is set during the winter of 1870-1871, several months after the fall of Napoléon III's Second Empire (early September 1870) but before the preliminary peace treaty signed in Versailles on 26 February 1871. Since the Prussian forces entered Rouen in December 1870 (apparently, Prussian soldiers even used Flaubert's bed), the siege of Paris (19 September 1870 - 28 January 1871) is still ongoing; the Government of National Defence (4 September 1870 - 13 February 1871) was in a sense the first government of the Troisième République.
The "Empire" is the "Second Empire", established after Louis-Napoléon Bonaparte's coup d'état of December 1851 and the man's coronation as Napoléon III in December 1852. His reign is divided into two periods: an authoritarian period (1852-1860) and a "liberal" period (1860-1870). Those who supported him, like Boule de Suif, were bonapartists; there were several types of opposition, such as Orléanism (supporting the constitutional monarchy by the House of Orléans, represented in the story by the comte Hubert de Bréville) and republicanism (represented in the story by Cornudet).
The Legion of Honour / Légion d'honneur was established in 1804 by consul Napoléon Bonaparte (months before his coronation as emperor) because the French Revolution has abolished all types of decorations that existed under the Ancien Régime. From the beginning, the Legion of Honour could be awarded both to servicemen and to civilians. Napoleon is supposed to have said, "Je veux décorer mes soldats et mes savants" ("I want to decorate my soldiers and my scholars").
The General Council refers to the "conseil général de département", a type of assembly established after the French revolution; since 1833, the members of this assembly were elected. During the Second Empire, the conseil départemental was a kind of deliberative assembly, since government had been centralised and the executive power at the level of the département was in the hands of the préfet de département, who represented the centralised power. However, during the liberal period of the Second Empire, there was some pressure to decentralise government and the conseil général was given a few more competencies. (See Histoire du conseil départemental du Loiret; I assume that the description there is representative of the Second Empire in general; and Le Conseil Général du Pas De Calais, which points out that the conseil départemental did not have much real power.)
The "well-disposed opposition" (there is no uppercase letter in the French original: "opposition bienveillante") is a bit harder to define, especially when taking into account hat the powers that be could could propose official candidates / "candidats officiels" for elections of assemblies such as the conseil général. Since Carré-Lamadon had been awarded the Légion d'honneur, it is likely that he was such an official candidate. The best explanation for his "opposition" that I can think of is that he may prefer the older more authoritarian style of government to the more liberal style that dominated since 1860. The entire description of his political position ("the well-disposed Opposition, merely in order to command a higher value for his devotion when he should rally to the cause which he meanwhile opposed" / "l’opposition bienveillante, uniquement pour se faire payer plus cher son ralliement à la cause qu’il combattait") suggest that he does not want to commit himself too strongly so he can turn his coat when the political winds should change; in other words, he is practising the venerable political art of opportunism.