If the top card of your library changes while you're casting a spell, playing a land, or activating an ability, you can't look at the new top card until you finish doing so. This means that if you cast the top card of your library, you can't look at the next one until you're done paying for that spell.

You must follow all normal timing rules for spells cast and lands played using the third ability and must pay all costs for spells cast this way. To cast a spell this way, you may choose to pay any alternative costs or optional additional costs the spell may have (such as a kicker cost). You must pay any mandatory additional costs.

The top card of your library isn't in your hand, so you can't cycle it, discard it, or activate any of its abilities that could be activated from your hand.

The Lunar Whale lets you look at the top card of your library whenever you want (with one restriction—see below), even if you don't have priority. This action doesn't use the stack. Knowing what that card is becomes part of the information you have access to, just like you can look at the cards in your hand.