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.

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.