I'll get the simple bit done first - you're right, nothing in the Thrown property turns a melee weapon into a ranged weapon. It's a melee weapon you can use to make ranged attacks, so the Archery Fighting Style, which says that

You gain a +2 bonus to attack rolls you make with ranged weapons.

can never apply to it. It can apply to darts, because they are thrown ranged weapons. This was also confirmed in a recent Sage Advice column:

Does the Archery fighting style work with a melee weapon that you throw? No, the Archery feature benefits ranged weapons. A melee weapon, such as a dagger or handaxe, is still a melee weapon when you make a ranged attack with it.

Now for the Dueling Fighting Style. Here, I have to disagree with you - when you roll damage for a thrown weapon, you aren't wielding that weapon in one hand. You were wielding it, then you threw it, then it hit. If you were still wielding it when it hit the target, it would be a melee attack. (Unless you threw it, then teleported across the battlefield and caught it right when it hit them, which sounds cool, but doesn't seem particularly useful.)

"Wielding" is a fairly nebulous term, so there is room for interpretation here, but personally I'm inclined to think that "wielding [...] in one hand" pretty clearly requires you to be holding something in your hand.

On the other hand, Crawford says that the Dueling Fighting Style does work with thrown melee weapons, and he's the authority on these matters.

The question is whether thrown melee weapons qualify for Dueling or not

The description specifies that you pull daggers out of the armband (despite the logistically problems of sticking a full-sized dagger in an armband. I would have gone with knives had I designed it), which are on the melee weapon table and therefore are melee weapons (magical ones, no less). You obviously don't have another weapon in your other hand, so all we need to know is whether throwing a melee weapons counts as "wielding" the weapon.

There is no game-definition for "wielding", so we rely on common English to understand it. Google gives this definition for "wield":

hold and use (a weapon or tool).

Under that definition, and in my own opinion, I'd say that throwing a melee weapon counts as wielding it for the purposes of Dueling.

Crawford agrees that Dueling applies to thrown weapons, for what that is worth (keeping in mind his Sage Advice tweets are unofficial).

The related question Does Dueling Fighting Style apply to thrown weapons? has arguments for and against this ruling.

This is a poorly worded item

It is unclear whether you are drawing both daggers at once in both hands, or one at a time, or what. For some interpretations it would matter whether you are drawing both at once, or if you are throwing both at once, but this is mine: I assume you are drawing one, throwing one, then repeating. If you draw one, immediately hurl it, then draw another and immediately hurl it, you've still drawn 2 daggers and immediately hurled them.