Do your fists count as a brawl weapon (assuming you have at last 1 rank of brawl) or do you have to have something like a vibró knuckler or shock gloves?

I suppose that's up for interpretation, but it does say have a weapon equipped , while simple brawling is called unarmed brawling. I would rule that you at least need a weapon, but that's just me.

F&D pg 141, Defensive Training

"When the character wields a Lightsaber weapon, Melee weapon, or Brawl weapon, that weapon gains ranks in the Defensive item quality equal to the character’s ranks in Defensive Training (this replaces any ranks in Defensive the weapon already has)."

You need a weapon, which receives the Item Quality.

Brass knuckles only cost 25 credits a piece, so it's not that hard to pick up some brawl weapons any way. I assume it's written this way as your intended to need something to use defensively, catching a Vibro Sword with your bare hands is not going to be easy, but having something in your hands to protect your skin makes it much easier.

Yeah, I'd say you'd need an actual Brawl weapon (such as brass knuckles) in order to gain the benefit of Defensive Training. Otherwise, Defensive Training starts to encroach on the "is this too good?" territory since it'd be an "always on" boost to melee defense, with the only other comparison being Superior Reflexes which is often found as a 25 XP talent, is non-ranked, and provides an unconditional +1 bonus to melee defense.