Astral Heron

Yes — assuming it's played to a battlefield.

Play, in a trigger condition, refers to a card resolving, not merely being placed on the chain.^{[350.1][359][359.2]} Units resolve the moment they finalize: becoming a game object, gaining their active abilities, and entering the board at the chosen location, all in that same moment.^{[359.2.a][359.2.c]}

A trigger's condition is evaluated after a potentially inciting event has been processed, and nothing requires that event to have been caused by a different card.^{[383.2.c][383.2.c.1]} The ability's text splits into two parts: "When you play your first card each turn" is the trigger condition, running up to the first comma,^[383.2.a] and "if I'm at a battlefield" is the first thing after that comma, making it an intervening if — checked at trigger time, not on resolution.^[383.3.e] Astral Heron's own resolution satisfies both parts at once: it's simultaneously an instance of "you play a card" and the moment it arrives at a battlefield. If it's the first card played this turn and it's played to a battlefield, both are satisfied at that same moment and the ability is placed on the chain.

Yes — the discount still applies to your next card.

Once Astral Heron's triggered ability is already a chain item, waiting to resolve, killing or moving Astral Heron in response to it changes nothing about how it resolves.

"If I'm at a battlefield" is an intervening if, so it's only checked once, at the moment the triggering event occurs, to decide whether the ability is placed on the chain at all — it is not re-checked when the ability later resolves.^[383.3.e] Once triggered, the ability is an independent item on the chain, and nothing in its effect references Astral Heron's location or continued presence, so there's no referent left for a later removal to invalidate.

So an opponent reacting to the trigger by killing, bouncing, or moving Astral Heron before it resolves doesn't stop the reduction: the trigger condition and intervening if were already satisfied the moment the ability went on the chain, and it resolves exactly as if nothing had happened.

No — the discount is spent the moment it reduces that spell's cost.

Astral Heron's triggered ability resolves into a delayed passive ability, active only for "your next card" — a single specified window, not a standing effect.^[391] That window belongs to finalizing the next card, not to that card resolving successfully — it doesn't linger to see whether the spell goes on to succeed. Discounts are applied while determining a card's total cost, in step 3 of the play process, and the reduced cost is paid in step 4 — both steps happen well before the card is even finalized, let alone placed on the chain where it could be countered.^{[356.4][356.4.a][357]} Whichever card is played next consumes that window the moment its cost is reduced; nothing that happens to that card afterward can reopen it.

A countered card does nothing and is not considered to have been played,^{[425.1.a][425.1.b]} and countering does not refund any costs paid to play it, additional costs included.^{[425.1.c][425.1.c.1]} That confirms the reduced cost itself stays spent — but Astral Heron's discount was already used up before the spell was even vulnerable to being countered, so there was never anything left to give to a different card.