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.

No, you can't take the -5/+10 while throwing a dagger.

Thrown: If a weapon has the thrown property, you can throw the weapon to make a ranged attack.

Throwing a dagger is a ranged weapon attack, so the first two benefits of the Sharpshooter feat will apply. However, the third benefit doesn't require a ranged weapon attack:

• Before you make an attack with a ranged weapon that you are proficient with, you can choose to take a -5 penalty to the attack roll. If the attack hits, you add +10 to the attack’s damage.

It specifically requires that you attack with a ranged weapon. The thrown property doesn't change the fact that a dagger is a melee weapon. Throwing a dart, for example, would benefit from the -5/+10, because it's a ranged weapon. The thrown property isn't what's important here - what's important is whether the weapon is a ranged weapon or not.

If you prefer to hear it from the designers, Jeremy Crafowrd agrees that the first two benefits of Sharpshooter work with daggers, but not the 3rd. He also stated that a dagger is a melee weapon, even when you throw it, as well as including it in the official Sage Advice Compendium:

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.