Rashmi, Eternities Crafter

Rashmi's ability triggers when you cast your first spell each turn, not just on your turn.

If you cast the top card of your library, you do so as part of the resolution of Rashmi's ability. You can't wait to cast it later in the turn. Timing permissions based on the card's type are ignored, but other restrictions (such as "Cast [this card] only during combat") are not.

If Rashmi's ability resolves and the spell that caused it to trigger has been countered, use that spell's mana value as it last existed on the stack to determine whether or not you may cast the top card of your library.

For spells with {X} in their mana costs, use the value chosen for X to determine the spell's mana value. For cards in your library with {X} in their mana costs, X is considered to be 0.

If you cast a card "without paying its mana cost," you can't choose to cast it for any alternative costs, such as emerge costs. You can, however, pay additional costs. If the card has any mandatory additional costs, such as that of Cathartic Reunion, those must be paid to cast the card.