1-Now, for the Dueling Fighting Style a Fighter can pick, can that +2 damage bonus apply a versatile weapon? Or is it restricted to only one hand, one weapon?
You can get the +2 damage bonus if you are "wielding a melee weapon in one hand and no other weapons" (PHB 72). That's fine with a longsword, but it means you aren't wielding the weapon in two hands, so you use the base damage, not the parenthetical damage entry next to the Versatile property.
In contrast, on that same page the Great Weapon Fighting fighting style triggers its ability, "for an attack you make with a melee weapon that you are wielding with two hands," and calls out wielding a weapon with the two-handed or versatile properties.
2-Assuming the DM allows the Human variant trait, I will be using it for my character. What feat would be the most worth picking up? I have my eyes on Great Weapon Master but, since a Long sword isn't a heavy weapon, I can't apply the "Power Attack" bonus to it so that sort of sucks.
None of the PHB feats have any optimal synergy with the longsword as a stand alone weapon. You can't use Defensive Duelist, Dual Wielder, Polearm Master, or Shield Master. Great Weapon Master is still solid, but as you noted you miss out of one of the feature. Savage Attacker, Lucky, Sentinel, and Martial Adept all give you an attack boost regardless of the weapon you wield.
Most of the feat specialties are appropriate for Samurai, though: melee attacks with a katana aren't the Samurai's bread and butter in an adventuring or wartime scenario. The battlefield is very different than a formal duel. In a battle the katana served as a badge, a back-up weapon, and the mean for collecting the heads of noteworthy opponents. In a war the Samurai made most of his kills using his best specialty among the other traditional weapons of the samurai - the great bow (daikyu), the glaive (naginata), or the great sword (no-dachi). There are fighting styles and feat options for each of these specialties and even the odd-ball who fights with both blades of his Daisho, but there's no similar specific support for a katana specialist.
Besides, what self respecting Samurai would want to risk his soul and sacred honor brawling with a rust monster or some acidic ooze pile? What kind of man sullies his grandfather's blade with the blood of bakemono? That's simply not how things are done in civilized society.
But if you have to be a barbarian about it and hit everything that crosses your path with your katana ... I suppose Martial Adept with Dueling Style and Battle Master will make your character distinctive enough from the alternatives. He'll have a significantly bigger bucket of butt-kicking maneuvers and superiority dice to use than another Battle Master Fighter - especially at the first half of his career. You even open up Maneuvers 2 levels ahead of any other Fighter, so maybe your character could be considered a Kensai ("Sword Saint").
Marty Lud