[RPG] Are you still two-weapon fighting if you’ve thrown one of your two weapons and are no longer holding it

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.

From the rule you quote:

If either weapon has the thrown property...

Which weapon(s)?

a light melee weapon... [and] a different light melee weapon.

So the weapons of "either weapon" mention are the light melee weapons from the previous paragraph.

You can't throw both darts, because they're not valid weapons for two-weapon fighting.

And neither is a melee weapon, so:

You can't make improvised (dagger?) melee attacks with both.

But doesn't using a dart as an improvised dagger make it a melee weapon? "Improvised Weapons" (PHB p.147-148) says "in many cases, an improvised weapon is similar to an actual weapon and can be treated as such." (emphasis mine)

Nope. Later in "Improvised Weapons" we have "if a character uses a ranged weapon to make a melee attack...." (emphasis mine) The authors could have said "ranged weapon as a melee weapon" or something that indicated some change in the weapon's type, but they didn't.

It's still a ranged weapon, per the table, even when used to make a melee attack.