I don't know the answer to your first question. For your second, it does not appear that Louallier got his habeas corpus. Apparently he was held in jail by Jackson even after being found innocent by a Court Marshal and was only freed after news of the end of the war reached New Orleans. I don't think Louallier was held in jail very long since it appears that the offense for which he was held occurred some two months after the Battle of New Orleans.
From an 1841 New Orleans City Directory:
The war with Great Britain followed immediately afterwards. With all its incidents, ending in the GLORIOUS 8th of JANUARY, our readers are familiar.
About two months after the British had retired, Louallier, a member of the Legislature, made a publication against an order of General Jackson, removing to one hundred and twenty miles beyond the city, a number of Frenchmen, who, after joining bravely in the defense of the country, had obtained certain certificates of their national character, from the French Consul, for the purpose of leaving the army, which was still kept under arms, and returning to their families.
This was immediately followed by a writ of habeas corpus, issued by Hall, the Judge of the United States court, for the release of Louallier. Whereupon, Judge Hall, in turn, was also imprisoned. Louallier was acquitted by a Court Martial; but neither he nor the Judge were released until official news of the peace had arrived.
Hall had no sooner gained his authority as Judge, than he punished Jackson for a contempt of Court, by fining him one thousand dollars, which was paid by the people of the City.
Years later, the US Congress paid Jackson's fine with interest. I wonder whether the people of New Orleans ever got their money back.
Marshal = Martial
DOW v. JOHNSON, 100 U.S. 158 (1879) (at 194-5)
It is impossible in discussing this matter that memory should fail to recall a very famous case of historical interest, involving many of the same principles, which occurred about half a century before this, and of which the same city was the theatre. During what has been called the siege of New Orleans, at the close of the last war with Great Britain, the commanding general of our forces declared martial law in that city. This was unpleasant to many citizens, and to others who claimed to be foreigners domiciled there at the time. Some of these becoming restive under its restraints, made publications of a seditious character in the newspapers, for which they were arrested by order of General Jackson. When Judge Hall, of the proper civil court, issued a habeas corpus for their release, the general tore up the writ and sent the judge by force beyond his lines. Within a very few days after this, the victory of the 8th of January, 1815, was achieved, and on the receipt of the news of the treaty of peace the declaration of martial law was revoked. Judge Hall, on resuming his judicial functions, issued a process against General Jackson for contempt of court in his action in reference to the writ of habeas corpus. That distinguished man, though in the midst of the adulation consequent on the great victory, did not act as the defendant in this case did, by paying no attention to the process, but came to the court in citizen's dress, attended only by a single member of his military family and with his legal adviser. He offered to read the same paper which his counsel had read against issuing the process for contempt, and when the court declined to hear it, submitted himself to its judgment. At this there was such a demonstration of ill-feeling in the crowded court-room that the judge said he could not proceed, and would adjourn the court. But the noble defender of the city declared that he was equally ready to defend the court, and begged that the judge would proceed without fear to do what he might think his duty required. A fine of $1,000 was entered up against the general, which he paid at once, and used his authority, which was needed, to disperse the mob, who were inclined to violence against the judge.
I confess I have always been taught to believe that Judge Hall was right in imposing the fine, and that General Jackson earned the brightest page in his history by paying it, and gracefully submitting to the judicial power. Such I believe is the judgment of history and of thoughtful judicial inquirers; though a grateful country very properly refunded to her favorite general the sum he had paid for a necessary but unauthorized exercise of military power. I have no doubt that General Dow had good reasons for all he did, and I think he would have acted more wisely if, respecting the courts in the proper exercise of their functions, he had made his defence at the right time before the appropriate tribunal.
In March 1815, while New Orleans was still under martial law, Louis Louillier was tried by a general court-martial for a number of alleged offenses, including spying. (8) The court-martial found it only had jurisdiction over the spying offense, of which Louillier was acquitted. (9)(8.) Winthrop, app. XII, American Articles of War of 1806, 56 and 57, 981; Ibid., 101, sec. 2, 985. Articles 104 (aiding the enemy), 106 (spies), and 106a (espionage) of the Uniform Code of Military Justice (UCMJ), enacted in 1950, track these early Articles of War provisions closely (10 USC, secs. 904, 906, 906a (2002); Winthrop, 822.
(9.) John Spencer Bassett, The Life of Andrew Jackson (New York: Macmillan, 1928), 226-27.