Can I kill Lurrus directly after it enters the battlefield, before it's controller gets priority to cast a permanent from their graveyard?

Question

The two following rulings for Lurrus of the Dream-Den apply:

You must follow the normal timing permissions and restrictions of the spell you cast from your graveyard.

Once you begin to cast the spell, losing control of Lurrus won’t affect the spell. You can finish casting it as normal.

By the second quoted rule, I see if I don't respond to Lurrus being cast and entering, and my opponent chooses to cast a permanent from their graveyard, even if I then kill Lurrus, the chosen permanent will still be cast.

So my question is if my opponent casts Lurrus and it enters the battlefield, can I immediately kill it before my opponent gets priority to cast a permanent from their graveyard.

Answer 1

Not normally. When ever a spell resolves, per the rules of magic, the active player gets priority.

117.3b The active player receives priority after a spell or ability (other than a mana ability) resolves.

This means that your opponent who cast Lurrus will almost always have the opportunity to cast a spell before you get priority to try and remove Lurrus, in the same way players pretty much always have the ability to use a planeswalker ability before their opponents can kill the planeswalker.

There are three edge cases I can think of that would make this untrue:

1. Lurrus entering the battlefield caused some ability to trigger, say Soul Warden. This means that something goes on the stack before Lurrus' caster has priority, and since the spells Lurrus allows to be cast are generally not instant speed, the caster will need to pass and try to resolve that effect first. You can respond to that triggered ability when you get priority with an instant meant to kill or exile Lurrus.
2. Lurrus was cast with flash and someone other than the caster is the active player. In this case Lurrus' caster still needs to wait for their turn to cast anything using Lurrus' ability and he can be dealt with. This might work if it was cast with flash on a non-empty stack during the caster's turn, but not likely as the permanents meant to make use of Lurrus would probably also be under the same effect.
3. Lurrus' caster is prevented from casting additional spells - there are about a half dozen spells that would prevent additional spells being cast with Lurrus, Rule of Law is the most recent one, Curse of Exhaustion would only affect one player, and Ethersworn Canonist would prevent casting anything other than an artifact after Lurrus, etc.

Answer 2

Unfortunately, the rules say

117.3b The active player receives priority after a spell or ability (other than a mana ability) resolves.

So if it is your opponent's turn, then once Lurrus has finished resolving and entered the battlefield, your opponent receives priority, and keeps it until they have declared that they will cast no more spells and activate no more abilities for now, passing priority to you.

In particular, if the stack is empty after Lurrus has resolved, they can cast a permanent spell before you are allowed cast anything.

The second quote you quote is actually rather rarely relevant. The process of casting a spell includes paying its costs. This cost can include sacrificing a creature, returning a creature you control to your hand, or similar. What the quote says is that you're allowed to pay such costs with Lurrus, and still complete the casting, even if Lurrus is the reason you were allowed to cast it in the first place.