Rainbow Lasso clarification

I just played second edition for the first time, and saw that Unicorn Lasso got replaced by Rainbow Lasso. In our game we came across an interesting situation.

A player had Rainbow Lasso in their stable and paid the price to activate the card (discarded 3 cards from their hand). They then stole Dark Angel Unicorn from my stable (if this unicorn enters your stable, you may sacrifice a unicorn card, then bring a unicorn from the discard into your stable). Could this player have taken a unicorn from the discard pile that was required for them to discard from the Rainbow Lasso?

All effects of a card happen simultaneously; does this extent to chain effects? (The 3 cards haven't entered the discard pile yet) Or is each chain of effects treated separately; this would be the third effect in the chain? (The 3 cards enter the discard pile after the first card is resolved)

Thanks! I'm out here forming plans for unicorn domination!