Probably not. Tavern Brawler requires you to "hit the creature" with a weapon attack. While the Disarm rule involves making an attack, it's not clear that that attack involves hitting a creature. Normally, hitting a creature requires beating its AC:
When you make an attack, your attack roll determines
whether the attack hits or misses. To make an attack roll,
roll a d20 and add the appropriate modifiers. If the total
of the roll plus modifiers equals or exceeds the target’s
Armor Class (AC), the attack hits. The AC of a character
is determined at character creation, whereas the AC of a
monster is in its stat block.
Since the attack you make to Disarm isn't against a creature's AC, it probably doesn't involve hitting a creature, and therefore can't trigger Tavern Brawler.
Yes.
As you noticed in the DMG, a disarm attempt is a hybrid attack/contest: attacker's attack roll against defender's athletics or acrobatics check.
As to your specific example, you've (basically) got it all right there, too. Let's step through it.
a lvl 1 Human Fighter with a shield and no other weapon could attack the defender with a shield (Tavern Brawler)
Yes. It'll be as an improvised weapon, but nicely for a tavern brawler he'll be proficient.
declare they are attempting a disarm before damage is rolled (atk vs skill)
I believe you would have to declare that the attack is a disarm attempt before the attack roll, not after the attack roll but before any damage, as this seems to imply. The reasoning: if you could see the attack roll, you know it doesn't beat the AC, then declare "oh, I was trying to disarm," then that stinks mightily of cheese.
and if successful disarm the defender
Yup, per "disarm" in the DMG,
then grapple as a bonus action
Yup, per tavern brawler,
and move the defender from their weapon by 15ft?
Yup, per "Grappling," PHB p.195 (assuming your base speed's 30').
Best Answer
Yes, you can
This is addressed explicitly in the Sage Advice Compendium:
Whether Stunning Strike or Disarm, if the requirement is a melee weapon attack, you can use unarmed strikes, as they count as a melee weapon attack. And a melee weapon attack is a weapon attack.