In a way, both North and South both built their armies from scratch. The "Regular Army" (the professional military that had existed in peacetime) consisted of roughly sixteen thousand men, most of which were stationed in scattered garrisons in the West, and a third of these officers resigned to serve the Confederacy. Lincoln called for 75,000 volunteers in the wake of Fort Sumter, while the Confederate Congress approved an army of 100,000 volunteers in March 1861. These men formed new units from towns and counties, electing their own officers (to mixed results, as these local bigwigs weren't always the best at military tactics. This practice died away around 1863 with the institution of officer review boards.) The Regular Army was kept intact by Winfield Scott; James McPherson describes this as a mistake, as "hundreds of officers and noncoms in the regular army could have provided drill instructors and tactical leadership to the volunteer regiments. But Scott kept them with the regulars, sometimes far away on the frontier, while raw volunteers bled and died under incompetent officers in Virginia." The South, with no regular army, had to build its own army from the "313 officers who resigned from the U.S. army ... [who] contributed a crucial leaven of initial leadership" along with the graduates of "eight military 'colleges,' [seven of which] were in the slave states."

As for the second part of your question, there were three others by the time Lincoln took office, which Shelby Foote identifies as "all Florida forts: Pickens off Pensacola Bay, Taylor at Key West, and Jefferson in the Dry Tortugas." The main reason why so few forts remained in Union hands lies in the hands of President Buchanan, who wavered in providing instruction to troops stationed in the South. McPherson notes, "Buchanan's forceful denial of the legality of disunion ended with a lame confession of impotence to do anything about it. Although the Constitution gave no state the right to withdraw, said the president, it also gave the national government no power 'to coerce a State into submission which is attempting to withdraw." For a concrete example, Foote mentions "[David E.] Twiggs, a Federal brigadier in command of the Texas Department during the secession furor, had repeatedly asked Washington for instructions through that stormy time. Receiving none, he acted in accordance with a statement he had made: 'If an old woman with a broomstick should come with full authority from the state of Texas to demand the public property, I would give it to her.' He did just that, surrendering all the troops, forts, and equipment in his charge, not to an old woman, but to a posse of citizens who styled themselves a 'committee for public safety.' Northern howls of 'treason to the flag' went up, and Twiggs, being summarily dismissed from the U.S. army, repaired forthwith to new Orleans, where he was solaced and rewarded with a commission as a Confederate major general."

Sources: James McPherson, The Battle Cry of Freedom (Oxford University Press, New York, 1988), Chapter 8: "The Counterrevolution of 1861"

Shelby Foote, The Civil War: A Narrative, Volume 1: Fort Sumter to Perryville (Vintage Books, New York, 1986)

By and large no, active regular army units did not switch sides, and in general a much smaller % of enlisted men went South than Officers.

The Regulars were also really just too small to make much difference, nowhere were they concentrated in anything larger than Battalion size, and even then mostly on the Plains.

As for in the South one notable remaining Fort was Fort Pickens, in Pensacola Florida. The Fort was on an island and controlled access to the Bay, and despite repeated threats and posturing the garrison held out, and were not actually attacked and remained with the Union through the war.

We also have Harpers Ferry which was one of the major armories of the federal govt before the war, and whose works and stocks were seized upon Virginia's secession.

On the other side, what happened to the military after the war? Did Confederate soldiers rejoin the United States army?

I'd love it if the answer included remarks on the Union's reputation for professional military tactics and the Confederate reputation for effective skirmishing and committed bayonet charges.