[RPG] Can Monks gain the benefit of the Dual Wielder Feat with their Unarmed Strikes

No. You can't dual wield Unarmed Strikes because they are not considered a Light weapon. That said, I don't believe it would be gamebreaking to houserule it that way.

However, in any case, you can't use two-weapon fighting and the Monk's martial arts on the same turn, because each uses a bonus action, and you only get one bonus action per turn. So your example of "tri-wielding" with a quarterstaff attack, an off-hand Unarmed Strike, and then a Martial Arts bonus Unarmed Strike would not be possible, because it would require two bonus actions to pull off.

Two weapon fighting:

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 your other hand.

Martial Arts:

When you use the Attack action with an unarmed strike or a monk weapon on your turn, you can make one unarmed strike as a bonus action.

Note that the Monk Martial Arts ability does allow the Monk to effectively dual wield Unarmed Strikes, since they can unarmed strike as their main Attack action, and as their bonus action. And unlike regular Two Weapon Fighting, they get the bonus to their damage roll on the bonus attack as well.

No.

Tavern Brawler's relevant ability is:

You are proficient with improvised weapons and unarmed strikes.

The rules for improvised weapons in the PHB say two things:

• Some non-weapon objects can just be treated as weapons if they are mostly similar to weapons.
• Other objects just do 1d4 damage.

That interacts with the proficiency rules. If an object is essentially just a club (table leg, wooden branch, torch), spear (pitchfork), greatclub (log, broken stalagmite), or whatever, then a character proficient in simple melee weapons is proficient in using such weapon-like objects as weapons.

Objects that "bear no resemblance to a weapon" (PHB, p. 148) aren't treated as weapons and as such characters can't be proficient with them. A table leg is close enough to a club for club proficiency to transfer; a candelabra or a serving platter or a horse's saddle ... not so much.

(As a side note, many tools that a character may be proficient with may also be treated as weapons in the above sense, but unless the character is proficient with their use as weapons, they can't use their proficiency bonus when attacking with them).

Tavern Brawler's effect is to give proficiency with anything being used as a weapon, even if it's not remotely weapon-like. That's not the same as being able to treat it like a weapon!

Dual wielder grants +1 AC if the character is wielding separate weapons in each hand. Proficiency isn't a factor. We can test various cases for Tavern Brawlers getting this bonus:

• No weapon in either hand. No Bonus. Unarmed strikes are not weapons.
• Club-like improvised weapon in both hands. +1 AC Bonus. This is the same as if the character had a "real" club in either hand.
• Unbalanced non-weapon object in one or both hands. No Bonus. They aren't weapons.
• Shield on one arm, club-like object in the other hand. No Bonus. A shield doesn't "bear any resemblance" to a weapon.

In fact, Tavern Brawler doesn't do anything that would change whether the Dual Wielder bonus applies. It doesn't say that you can treat non-weapon objects as if they were weapons; it just lets you use your proficiency bonus with them.

As far as attacking with shields goes, it's worth noting that the Shield Master feat doesn't give any enhanced ability to do damage with a shield, just to shove and block with it.