My question is this: if Larfleeze has the Crimson Gem of Cytorrak and makes a close combat attack through an Orange Construct token against a character to which he himself is not adjacent to when holding an object, does he still get the damage from the object?

Bonus Question: If he does make this attack, and the object can be used, and the object is the Dumpster, do you drop it next to the character adjacent to the Construct token or next to Larfleeze?

All relevant text follows:

ORANGE CONSTRUCTS: Orange Construct objects are objects that can't be picked up and are removed from the game if Larfleeze is KO'd. Whenever Larfleeze is moved, after actions resolve you may move any or all Orange Construct objects an equal amount. Larfleeze considers characters adjacent to or in the same square as an Orange Construct object to be adjacent to Larfleeze for Poison and close combat purposes.

Crimson Gem of Cyttorak
This object can't be destroyed.
Once per game per character, give a character occupying this square a power action and roll a d6 that can't be rerolled. On a 5 or 6, place this object on that character's card. Modify that character's damage value by +1 and that character can use Charge, Super Strength, and Toughness. (Characters can't use more than 1 damage reduction power.) When that character is defeated, place this object in the square they last occupied. At the end of the game, if an opponent's character has this object on their card or all of your characters have been defeated, your opponent scores this object.

Dumpster
Heavy: This object is not destroyed when used in an attack. The attacker places it in any square adjacent to the target after use.

I would say "yes" he does, he is still doing the attacking.

I'd say you follow the rules and place the dumpster next to the target. Yes, it's wacky but you are following the literal interpretation of the rules.

Seems pretty straightforward that he can use objects in close combat attacks through his constructs, even if it makes no sense whatsoever.

And if it's a dumpster, the text of the dumpster clearly states a square adjacent to the target.

Maybe Larfleeze has a portal gun or something.

You should always follow a literal interpretation of the rules, and if you tweak them to "make more sense" or something like that it should be clearly stated as house rules.