When do you declare to pay the additional cost of cards? what is the order of when the opponent can react

You declare additional costs as part of playing the card itself, before any reaction window. The choice to pay an optional cost (like Accelerate) is made in Step 2 of the play process, that cost is added to the Total Cost in Step 3 (Rule 356.2), and you pay it — along with base costs and any mandatory "as an additional cost" costs — in Step 4 (Rule 357.1–357.2). The entire play process (declare → choose → pay → legality → finalize) completes before your opponent gets any priority; finalizing doesn't pass priority (Rule 337.1.a).

After the card is finalized, what happens depends on type: a spell stays on the Chain, its controller gets priority first and then priority passes to the opponent in turn order, who may respond with Reactions (only Reactions are legal in a Closed State). If the spell is countered, the costs you already paid are not refunded (Rule 425.1.c). Units and Gear instead resolve immediately on finalization (Rule 337.2), so there is no reaction window to the card entering play itself — opponents can only respond to "when you play me" triggers placed on the Chain afterward.

One caveat: costs written inside an effect ("Pay any amount of [A] to deal damage," e.g. Bullet Time) are not play costs — they're paid at resolution, so they don't need to be paid if the card is countered (FAQ #5229).