[RPG] Does the Thrown property mean I can attack with the DEX

Per Crawford, No: you can't use the paladin's Divine Smite with a thrown weapon.

The problem here, as you have correctly identified, is that the language is ambiguous: does "melee weapon attack" mean "a melee attack with a weapon" (your option 1), or "an attack with a melee weapon" (your option 2)?

Helpfully, rules designer Jeremy Crawford has answered this exact question in a tweet from August 2015 (which is not an official ruling), and in the following Sage Advice Compendium question on p10 of the v1.14 PDF (which is an official ruling):

What does “melee weapon attack” mean: a melee attack with a weapon or an attack with a melee weapon?

It means a melee attack with a weapon. Similarly, “ranged weapon attack” means a ranged attack with a weapon. Some attacks count as a melee or ranged weapon attack even if a weapon isn’t involved, as specified in the text of those attacks. For example, an unarmed strike counts as a melee weapon attack, even though the attacker’s body isn’t considered a weapon.

As such, we should interpret Divine Smite's trigger to mean "...when you hit a creature with a melee attack with a weapon...", which means that you can't use Divine Smite with a thrown weapon, as it isn't a melee attack.

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.