With there being multiple answers already, I feel a little weird about posting one of my own... but if it's an essay needed to be convincing, then an essay there shall be.
No, Player B Cannot Tribute Crooked Cook In This Scenario
The cornerstone of the confusion, as I understand it, is a false (albeit commonly made) equivalency between certain cards which handle this sort of interaction differently from one another. Specifically, the cards The Monarchs Stormforth and Lava Golem are being regarded as "doing the same thing," when in fact there are fundamental separations between the two. These separations - which unfortunately are more nuanced than they ought to be, given the prevalence of the Kaijus - are the tools we will use to evaluate the problem at hand. Let us investigate.
There is already an existing ruling for The Monarchs Stormforth, regarding Forbidden Lance:
A monster that has been affected by the effect of "Forbidden Lance" cannot be Tributed for a Tribute Summon using the effect of "The Monarchs Stormforth".
The question further raised here is, to quote OP,
Ok, but then it states that:
This card can be activated even if your opponent does not control any monsters.
This effect is not only applied to monsters your opponent controls when the effect resolves.
Is there a discrepancy here? Well, no - not really. Let us examine the effect of The Monarchs Stormforth:
Once during this turn, if you would Tribute a monster for a Tribute Summon, you can Tribute 1 monster your opponent controls even though you do not control it.
It's important to recognize here that this is not an effect being applied to monsters on the field at resolution - this is an effect that gives the player an additional "thing" to do for the duration of the turn. The distinction, of course, being that any monsters summoned after the resolution of The Monarchs Stormforth are also legal tributes by its effect.
Here's the rub: even though this isn't an effect being applied to the tributed monster at the time of resolution, it's still the effect of a spell card. Any tributes of the opponent's monster by the effect of The Monarchs Stormforth are exactly that: Tributes by the effect of The Monarchs Stormforth. So, if a monster is unaffected by Spell Card effects, then it cannot be tributed by the effect of a Spell Card. By this reasoning alone, we have enough to see why Crooked Cook cannot be tributed in the given scenario - while Crooked Cook is unaffected by other card effects, it cannot be tributed by the effect of Spell cards - like The Monarchs Stormforth.
Now, while we have technically answered the original question, it would be worthwhile to address the "part 2" of the question that came up after the fact: "So why can monsters like 'Lava Golem' tribute unaffected cards?"
This is a detail that eludes the understanding of many players, across all skill levels - yet it is an extraordinarily important concept to grasp:
Lava Golem can tribute unaffected cards because it is not using an effect to do so.
Understandably, this throws a large number of people for a loop. We can show that the summon of Lava Golem, in fact, is not a monster effect by trying to determine which class of monster effect it would be classified under - and subsequently recognize that it meets the criteria for none of them.
According to Konami's Problem-Solving Card Text rules,
All card effects that make a Chain Link include a colon " : " or a semicolon " ; " somewhere in their text (possibly both). If a card effect does not use either, it does not make a Chain Link.
So, we know that the summon of Lava Golem does not create a chain, and thus cannot be an activated effect. Because it is not an activated effect, it immediately rules out all but one of the types of monster effects according to the Official Rulebook - all that remains is for Lava Golem's summon to be by a Continuous Effect, except...
[A Continous Effect] is active while the Effect Monster Card is face-up on
the field. The effect starts when the face-up monster appears
on the field, and ends once that monster is gone or is no longer
Obviously, this cannot be the case - Lava Golem is in our hand when we go to summon it, so it doesn't meet the definition of a continuous effect. We have thus shown that Lava Golem's summon is not considered to be the application of a monster effect. So, what is it?
Lava Golem, and cards like it (e.g. the Kaijus, Quickdraw Synchron, Santa Claws, Spell Striker, &c) have a Summoning Procedure that the player carries out.
To my knowledge, there is no explicit documentation from Konami on this matter, so I can offer no official explanation. However, the concept is: the monsters tributed to summon Lava Golem are not tributed by a monster effect. The Spell banished from the GY to summon Spell Striker is not banished by a monster effect. The card sent from the hand to the GY to summon Quickdraw Synchron is not sent by a monster effect.
The cards in these scenarios are tributed/banished/sent to GY by the player, not by any card or effect, in exactly the same respect as Tuners/Non-Tuners are sent to GY for a Synchro Summon, or Link materials for a Link Summon, or "contact fusion" materials for a "contact fusion". It is not a card that is considered by the game state to be doing anything - it is you, the player. And because these methods are not considered to be card effects, no protection against them is offered to any card made unaffected by other card effects.
Hope this clears up any remaining confusion.