So far so good.
"Now if are to believe the case you are defending that the Southern States were merely in "rebellion", that would mean that they were still part of the Union, and that there were only lawless areas within them."
Not exactly. In many of the states there were no functioning aspects of the Federal government, due to criminal, insurrectionist activity. All lawful government had broken down.
"So if Linkum (sic) had the "authority" to free slaves in one part of the Union, why would he need more authority to free them in the rest of the nation?"
Sorry, you were doing so well too.
Lincoln's authority to emancipate slaves in the rebellious areas stemmed from his position as commander-in-chief. He acted to deprive the insurrectionists with as much slave labor as possible, so as to injure their socio-economic infrastructure. Emancipation was a war-related necessity.
Where the rule of law prevailed, in the loyal states and in those part of the rebellious states that were under federal control, the existing laws concerning slavery were followed. That is why slavery had to be abolished by the 13th Amendment rather than by executive order or congressional legislation.
A dictator might have tried to abolish slavery by fiat, but Lincoln instead worked within the constitutional system.
Amendment V: 'No person shall ... be deprived of life, liberty, or property, without due process of law; nor shall private property be taken for public use, without just compensation.'
There are, without doubt, occasions in which private property may lawfully be taken possession of or destroyed to prevent it from falling into the hands of the public enemy; and also where a military officer, charged with a particular duty, may impress private property into the public service or take it for public use. Unquestionably, in such cases, the government is bound to make full compensation to the owner.
Justice Taney, Mitchell v. Harmony, 54 How. 115 (1851)
Just exactly HOW did Lincoln abide by the Constitution?
Twaddle. It was a personal political goal, nurtured for years. It was also, in its timing, a propaganda move intended to deprive the South of European support after Antietam, and to paper over the ill-will of the Trent affair.