Edgewall Innkeeper's ability resolves before the spell that caused it to trigger. It resolves even if that spell is countered.

An effect may refer to a card, spell, or permanent that "has an Adventure." This refers to a card, spell, or permanent that has an adventurer card's set of alternative characteristics, even if they're not being used and even if that card was never cast as an Adventure.

If an effect refers to a card, spell, or permanent that has an Adventure, it won't find an instant or sorcery spell on the stack that's been cast as an Adventure.