Any spell you cast that doesn't have the type creature will cause prowess to trigger. If a spell has multiple types, and one of those types is creature (such as an artifact creature), casting it won't cause prowess to trigger. Playing a land also won't cause prowess to trigger.
Prowess goes on the stack on top of the spell that caused it to trigger. It will resolve before that spell.
Once it triggers, prowess isn't connected to the spell that caused it to trigger. If that spell is countered, prowess will still resolve.
You draw a card and discard a card all while Jeskai Elder's ability is resolving. Nothing can happen between the two, and no player may choose to take actions.