Astral Heron

Astral Heron triggers after your first card to finish resolving each turn, provided Astral Heron is at a battlefield then. If its triggered ability resolves, it discounts the next card you play this turn after that resolution. In a trigger condition, "When you play" is checked only when the act of playing that card is completed by its resolution.^420.4.a If a spell is countered instead of resolving, it does not trigger Astral Heron, and the first card you later resolve that turn can trigger it instead.^420.4.a.1

When the condition is fulfilled, the triggered ability is placed on the chain and must resolve before the discount exists.^{383.3407.5391.4} Once it resolves, "the next card you play this turn" means the next card you place on the chain and queue for finalization.^420.5 Astral Heron's discount is applied while determining that card's total cost, before you pay the cost and finish finalizing it.^356.4357

Yes, if Astral Heron is your first card played that turn and enters a battlefield as it resolves.

A unit resolves immediately after finalizing.^337.2 During that resolution, it enters the board at its chosen location.^359.2.c Only after Astral Heron finishes resolving is it considered played for trigger conditions.^420.4.a Triggered abilities of permanents can have their conditions evaluated while their source is on the board.^384.2 Astral Heron is therefore at the battlefield when the game evaluates its trigger condition after the completed play.^383.2.c

The immediately following "if I'm at a battlefield" is part of the trigger condition and is checked at that time.^{383.2.a383.2.a.1}

No, if the spell leaves Astral Heron at base when it finishes resolving.

A card is not considered played, and abilities that trigger from playing it are not evaluated, until that card finishes resolving.^350.1420.4.a Astral Heron's trigger condition is therefore checked only after the spell's movement instructions have been completed.^383.2.c Because "if I'm at a battlefield" is part of that trigger condition, the ability is not placed on the chain if Astral Heron is at base by then.^383.2.a.1

This differs from moving Astral Heron after its triggered ability has already been placed on the chain, when the condition has already been satisfied.^383.2.a.1

No. Once the triggered ability is on the chain, it still creates the discount even if you kill or move Astral Heron.

"If I'm at a battlefield" is part of the trigger condition, so it is not checked again when the ability resolves.^383.2.a.1 The effect does not need Astral Heron's location or continued presence. When the triggered ability resolves, it creates a delayed ability that is not associated with Astral Heron and remains applicable at its specified time even if Astral Heron has left the board.^391.4393

No. Once the discount has reduced the next spell's cost, countering that spell does not transfer the discount to another card.

Astral Heron's triggered ability creates a delayed passive ability that applies only during the specified window for "the next card you play this turn."^391.4392 The discount applies while determining that card's total cost, and the reduced cost is then paid before the card finishes finalizing.^356.4357 By the time the finalized spell can be countered, that specified window has ended.^359.3.c