The Stone Disk, from The Longest Journey, a game I haven't played in a looooong time
The Title "I've been all around Greece's paradise?" and hint 2:
"Greece's paradise" is Arcadia (the game's alternate world of magic) and "Iron Man's world" is Stark (the game's name for our world in the future). (Also, "around" was a clumsy hint at the object's shape.)
My pieces are associated with earth, sea and sky, dark, and time.
This refers from whom you obtain the pieces:
Earth: The Banda (diggers).
Sea and sky: A piece that was further split between the Maerum (mermaid-like creatures) and the Alatien (winged creatures).
Dark: The Dark People.
Time: The Venar exist "outside of time".
My adornments come from a source of etymology: a vessel, a mother, a mentor, and a power-hungry traitor.
Refers to the jewels.
In game, the etymology is that "Draic Kin" (the "correct" term for these beings) is the origin of the word "dragon".
Red -> Cortez -> Mentor
White -> Mother
Green -> Jacob McAllen -> Power-hungry
Blue -> Old God -> Vessel: Part of the game's lore is that the ancestors of the Maerum and the Alatien travelled to the precursor to Stark+Arcadia inside the Old God.
The vessel, the dark, and a foul mouth point the way to where I'm useful.
This refers to characters that guide you. The Old God instructs the Dark People to give the protagonist a star chart pointing the way to the entrance to the Guardian's Realm, which they originally made based on his supernatural communications. Then, the map is deciphered by Burns Flipper, a hacker who swears a lot (and as the answerer remembered, he uses a hover chair): "I remember a guy who looked like he was in a wheelchair or something (maybe just sat around a lot) who was vulgar...one of the first f-bombs I remember hearing in my life was an improvised middle name for the protagonist..."
Hint 3 is obviously:
A hint as to the name (and adventure game genre) of the game.
I must say,
I haven't played this game since I was a kid, so a bit (a lot) of research had to be done. But when your hint mentioned a female protagonist, I eventually thought of April Ryan...an oddly memorable name, I guess (also fits hint 1!).