With Killing Forge of the Future, should the target itself, since "kill" is a cost, before you start selecting targets, Forge should be already in the trash. Is this correct?

No, that is incorrect. You cannot target Forge of the Future with its own recycle ability.

According to the rules of Riftbound and confirmed by RiftJudge FAQ #10944, you must select all targets for an activated ability at the moment you finalize the ability on the chain.

Here is the breakdown of why this doesn't work:

1. Cost vs. Effect: The text "Kill this:" is an activated ability where killing the gear is the activation cost (Rule 150.1, FAQ #10759).
2. Targeting Finalization: You must choose your targets (the cards in trashes to be recycled) at the moment you activate the ability and place it on the chain (Rule 355.8, FAQ #10944).
3. The Timing Mismatch: At the exact moment you are choosing those targets, Forge of the Future is still on the board. You have not yet paid the cost to move it to the trash. Because it is not in the trash when targets are locked in, it is not a valid target for its own recycle effect.
4. Payment: Once the ability is on the chain with its targets selected, you then pay the cost by moving Forge of the Future to the trash. By the time the card reaches the trash, the targets for the ability are already locked in and cannot be changed.

In short, the card must already be in the trash to be a valid target for the recycle effect, but it cannot move itself to the trash until after you have already locked in the targets for that effect.