2021-11-19 : A cleave cost is an alternative cost that's paid instead of the spell's mana cost. Casting a spell for its cleave cost doesn't change the spell's mana value.

2021-11-19 : Exiling Alchemist's Gambit happens as part of its resolution. If the spell is countered, it will go to the graveyard as normal.

2021-11-19 : If an effect allows you to “cast a spell without paying its mana cost,” you can't cast that spell for its cleave cost.

2021-11-19 : If multiple “extra turn” effects resolve in the same turn, the most recently created turn will be taken first.

2021-11-19 : If you cast a spell for its cleave cost, that spell doesn't have any of the text in square brackets while it's on the stack.

2021-11-19 : If you somehow skip the extra turn Alchemist's Gambit gives you or skip that turn's end step, the delayed triggered ability never triggers. (If you cast Alchemist's Gambit by paying its cleave cost, the delayed triggered ability is never created at all.)

2021-11-19 : You can't cast a spell for both its cleave cost and another alternative cost. For example, if an effect gives an Alchemist's Retrieval in your graveyard a flashback cost of {U}, you can't cast it from your graveyard for its cleave cost.