Cost: 1 Armor of Doom
Concealed-Optional
Equip only to Dr. Doom.

KO Armor of Doom -> Equipped character can't be stunned this attack.

I am trying to understand if this equipment is any good. I remember that cards like Kooey Kooey Kooey were really good but its text checked your resources after it resolved.

Cost: 2 Kooey Kooey Kooey
KO Kooey Kooey Kooey -> If you control four or fewer resources and a JLI character, search your deck for a card and put it into your hand.

Doom will no longer be equipped when you KO this. Doom gets no effect? Or is it referring to the previously equipped character. Or do you get to save another character that happens to be equipped?

How exactly is this used, and how or why would anyone play this card?

The reference "equipped character" only clarifies that the character who had the Armor attached when the effect was played is the one who is getting that effect. The Dr. Doom who wore the Armor when you KO'd it for its effect is the one that gets the effect. So it does work as printed.

As to how good it is, I'll point out that it does cost a resource point, but it does mean it takes out-of-combat removal or multiple attacks to knock down one of your Dooms for the effort.

it's a kind of nimrod counter

It falls back that people playing Common Enemy used to play a 2-cost equipment (Kevlar) that did the same thing, and sometimes the correct play on 5 WAS to equip doom and boris a few times. This simply frees up 1 resource point for you if you're playing a doom-like deck; or, if old-school Equip Common Enemy was remotely viable in Golden, let's you use Stretch on 5 to equip Dr. Doom for free.