If you cast a spell that was countered, you can't cast another spell during the same turn.

Archon of Emeria looks at the entire turn to see if a player has cast a spell, even if Archon of Emeria wasn't on the battlefield when that spell was cast. Notably, you can't cast Archon of Emeria and then cast another spell during the same turn.

If an effect says that a land enters tapped unless a condition is met or a cost is paid, that land enters tapped even if that condition is met or that cost is paid if it's affected by Archon of Emeria's second ability.