My friend has Giant Killer out on his battlefield. I have Warstorm Surge and I bring out Gisela, Blade of Goldnight. Does Giant Killer get destroyed first as a creature before my friend can cast it's instant adventure? Or is he able to cast its ability to destroy Gisela and exile itself before being destroyed?
You seem to be misunderstanding how Adventure cards work.
Once Giant Killer is on the battlefield; the Chop Down instant/adventure does not do anything. Adventure allows you to cast the Adventure part from your hand like a normal instant/sorcery. When that spell resolves, it is exiled, and while it is exiled, you can cast the creature from exile.
715.3. As a player casts an adventurer card, the player chooses whether they cast the card normally or as an Adventure.
715.3d Instead of putting a spell that was cast as an Adventure into its owner’s graveyard as it resolves, its controller exiles it. For as long as that card remains exiled, that player may cast it. It can’t be cast as an Adventure this way, although other effects that allow a player to cast it may allow a player to cast it as an Adventure.
So your friend cannot do anything with Giant Killer's adventure; your Warstorm Surge triggers when you play Gisela, and if you choose to target Giant Killer, it will take 10 damage (5 which is doubled by Gisela) when the triggered ability resolves.
From the rules:
- Handling Triggered Abilities 603.1. Triggered abilities have a trigger condition and an effect. They are written as “[When/Whenever/At] [trigger condition or event], [effect]. [Instructions (if any).]” 603.1a
603.2. Whenever a game event or game state matches a triggered ability’s trigger event, that ability automatically triggers. The ability doesn’t do anything at this point.
603.3. Once an ability has triggered, its controller puts it on the stack as an object that’s not a card the next time a player would receive priority.
So Warstorm Surge's ability is a triggered ability, and it used the stack. While it's on the stack, your opponent can cast instants. However, Adventure is an alternative way that a card can be cast. Once Giant Killer is cast and on the battlefield, the Adventure is no longer available.
Also, there is no "his" battlefield. There is only one battlefield. Cards on the battlefield have different controllers, but are on the same battlefield.