Do I pay rampage cost on resolution or on putting on the chain? Can I pay optional cost after my opponents reactions?

You pay Rampage's [C] when you play it, not on resolution — and no, you can't pay it after your opponent's reactions.

Rampage's text is "As you play this, you may pay [C] as an additional cost," which makes it an Optional Additional Cost (Rule 356.2.b). You declare it in Step 2 (Choose), it's added to the Total Cost in Step 3, and you actually pay it in Step 4 (Rule 357.2). The whole play process finishes — and the spell is finalized on the chain — before your opponent ever gets priority to react, so the decision and payment are already locked in. If they counter Rampage, the [C] isn't refunded (CR 426.1.c).

This is different from "cost within instructions" templates like Bullet Time's "pay any amount of [A]," which are paid during resolution and can be skipped if the spell is countered (CR 204.3.a). Rampage doesn't use that template, so it can't be paid late.