Does unarmed attack act the same as duel wielding? So you'd get your regular attacks and 1 extra as a bonus action?

Do you still lose your ability mod to damage for your "off-hand" attack?

Can these attacks be considered finesse attacks so you get to pick between Dex or Str for Attack and Dmg bonus?

So if you'd have the fighter style: Unarmed Fighting, with 2 attacks / round and Str score 20: (1d8+5) X 3 each combat round since you'd be able to attack twice with one extra as a bonus action. All three attacks still get your Str bonus to damage.

Fighter Style: Unarmed Fighting

Your unarmed strikes can deal bludgeoning damage equal to 1d6 + your Strength modifier on a hit. If you aren’t wielding any weapons or a shield when you make the attack roll, the d6 becomes a d8.

At the start of each of your turns, you can deal 1d4 bludgeoning damage to one creature grappled by you.

To duel weld you need to hold and use two different two melee weapons (one in each hand). There is a furthur restriction that they must be light unless you have the del wielder feat. Even with the feat you can not make an unarmed attach as a bonus action unless you are a monk.

Two-Weapon Fighting - PHB base details

When you take the Attack action and attack with a light melee weapon that you’re holding in one hand, you can use a bonus action to attack with a different light melee weapon that you’re holding in the other hand. You don’t add your ability modifier to the damage of the bonus attack, unless that modifier is negative.

If either weapon has the thrown property, you can throw the weapon, instead of making a melee attack with it.

Dual Wielder - PHB Feat

You master fighting with two weapons, gaining the following benefits:

• You gain a +1 bonus to AC while you are wielding a separate melee weapon in each hand.
• You can use two-weapon fighting even when the one-handed melee weapons you are wielding aren’t light.
• You can draw or stow two one-handed weapons when you would normally be able to draw or stow only one.

Unarmed Fighting - TCOE Fighter Fighting Style

Your unarmed strikes can deal bludgeoning damage equal to 1d6 + your Strength modifier on a hit. If you aren’t wielding any weapons or a shield when you make the attack roll, the d6 becomes a d8.

At the start of each of your turns, you can deal 1d4 bludgeoning damage to one creature grappled by you.

Melee Attacks - PHB BASE RULE

 

Instead of using a weapon to make a melee weapon attack, you can use an unarmed strike: a punch, kick, head-butt, or similar forceful blow (none of which count as weapons). On a hit, an unarmed strike deals bludgeoning damage equal to 1 + your Strength modifier. You are proficient with your unarmed strikes.

 
Those are all the rules above but the most important one is from PHB where it notes that Unarmed strikes from punch, kick, head-butt, or similar forceful blow do NOT count as weapons.

The Two-Weapon fighting rule is the next most important as It stipulates that to make a 2nd attack with your bonus action you must have a different melee weapon in each hand.

Investigating the Dual Wielder feat.  This removes the requirement for the weapons to be "Light" but is still specific about the weapons being "one-handed melee weapons"

Finally we check the "Unarmed Fighting" fighting style from TCoE but it specifically calls the attacks you deal with it's rules "unarmed strikes".  While it increases the damage and increases it even more if your hands are completely free of weapons and shields it does not specify that your fists or unarmed strikes are now regarded as weapons.

Ultimately, the only way I can think to gain the ability to make an unarmed strike as a bonus action is to take one level in Monk for the Martial Arts class feature which will grant unarmed strikes as bonus actions.

NOTE: the Unarmed Fighting style specifically mentions that you must not be "wielding any weapons or a shield" when you make unarmed strikes for them to do d8 damage.  It makes no mention of you potentially holding other objects, HOWEVER the "Improvised Weapon" rule would likely allow a DM to rule a good many things as counting to fulfill the "weapon" specification.

As others have stated Two-Weapon Fighting cannot be used with unarmed strike even with Dual Welder feat because it's not a weapon. Multiple official rulings in Sage Advice Compendium confirms that,, which eventually led to errata of the Player's Handbook for unarmed strike;

Can a paladin use Divine Smite when they hit using an unarmed strike? No. Divine Smite isn’t intended to work with unarmed strikes. Divine Smite does work with a melee weapon attack, and an unarmed strike can be used to make such an attack. But the text of Divine Smite also refers to the “weapon’s damage,” and an unarmed strike isn’t a weapon. If a DM decides to override this rule, no imbalance is created. Tying Divine Smite to weapons was a thematic choice on our part—paladins being traditionally associated with weapons. It was not a game balance choice

Does the Savage Attacker feat work with unarmed strikes? No. Savage Attacker relies on a weapon’s damage dice, and an unarmed strike isn’t a weapon (a point that was clarified in the Player’s Handbook errata).

What does “melee weapon attack” mean: a melee attack with a weapon or an attack with a melee weapon? It means a melee attack with a weapon. Similarly, “ranged weapon attack” means a ranged attack with a weapon. Some attacks count as a melee or ranged weapon attack even if a weapon isn’t involved, as specified in the text of those attacks. For example, an unarmed strike counts as a melee weapon attack, even though the attacker’s body isn’t considered a weapon. Here’s a bit of wording minutia: we would write “melee-weapon attack” (with a hyphen) if we meant an attack with a melee weapon.