Are we sure that they were actually CONSTRUCTS and not just fabricated or teleported?
Yes, they were constructs.
Oblivion was the embodiment of Kyle's bottled up negative emotions after his girlfriend was refrigerated, based on a comic book villain he made up as a kid, given life when Kyle was in the process of being possessed by Ion/absorbing the Central Battery energy from Halallax/given access to the Starheart by Jade.
When confronting Kyle, he commented words to the effect of "With apologies to Kermit the Frog, it's not easy not being green," with the other green lanterns being likewise given life from Kyle's subconscious. Alexandra De Witt was his love, Emerald Knight his heroism, the twins his hope for the future, Green Lightning his friendship with Wally, the manhunter green lantern his inner sci fi geek, etc.
It ends with him absorbing them all back into his mind where they belong.
And in the Lord Molvalio storyline, the entire solar system disappears after Hal Jordan takes his ring and flies off, which shouldn't have happened if they were fabricated by the ring or teleported.
The green lantern wiki assumes making constructs of other colors comes from light manipulation. Hal Jordan would make his ring invisible so it wouldn't be obvious he was green lantern without disarming himself I understand, and turned himself invisible too while fighting the Weaponers of Qward. According to the wiki Hal eventually gained enough finesse to make non green illusions.