[RPG] throw a longsword at someone

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.

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.