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.

No

But the argument is not so straightforward.

The problem with this question is that the RAW are murky enough to allow for some leeway in interpretation. Going by RAI, the intention was clearly to use Two-Weapon Fighting with one-handed weapons, and not game the system by attacking with a versatile weapon while holding it in two hands, arguing that it still counts as a one-handed weapon for the Dual Wielder feat.

The Rules

Dual Wielder

You can use two-weapon fighting even when the one-handed melee weapons you are wielding aren’t light.

While this qualifies the longsword for two-weapon fighting, the Versatile property states that:

Versatile

This weapon can be used with one or two hands. A damage value in parentheses appears with the property—the damage when the weapon is used with two hands to make a melee attack.

This explicitly states the need to use both hands to make a two-handed attack and use the bigger weapon dice, thus disqualifying this use case from the Dual Wielder requirement (one-handed melee weapons).

The Conclusion

While you can use a thrown handaxe and then attack with the longsword, you can only do so using the one-handed dice the Versatile weapon tag provides.