[RPG] Does the Tavern Brawler Feat mean you have to choose between Shield or Duelist

No, a monk can't use their Martial Arts feature with improvised weapons. Monk weapons are:

short swords and any simple melee weapons that don’t have the two-handed or heavy property.

Unfortunately, simple melee weapons are weapons that are in the "Simple Melee Weapons" category on the weapons table, and improvised weapons aren't present there. (Otherwise everyone who was proficient with simple melee weapons would be proficient with improvised weapons.)

So improvised weapons aren't simple melee weapons, and therefore they aren't monk weapons either and can't be used with Martial Arts.

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.