A surprised monster who launches an initiative of more than 20 years can use actions of air because he is able to act, and their turn is past
As you mentioned, the rules on air shares are as follows:
On the initiative counts 20 (loss of all initiative links), he can use one of his options for action. He can not do it as long as he is unable or unable to take action. If she is surprised, she can only use it after her first round of combat.
So, we know how to determine if a monster can take countermeasures:
1. Can he take action in general?
2. Is it surprised? If so, has his turn already passed?
If the answer to 1 is "no", then the monster can not do any action.
If the answer to 2 is "yes" then "no", then the monster can not take action.
The "Surprise" section says:
(…) If you are surprised, you can not move or act during your first round of combat and you can not react before the end of this round (…)
You are only prevented from taking action during your first roundSo, if it's not your turn, you will not be prevented from taking action.
If the monster scores greater than 20, count 20 for his initiative already past and therefore no method to prevent the actions of den is at stake. Note that in case they make The shares of Lair 20 will still be after their turn because they will "lose all ties", so they will not be prevented from taking shares of the lair if they cast more than 20 points.
Note that the last sentence of the quote is not redundant:
If she is surprised, she can only use it after her first round of combat.
If a monster is surprised and falls below 20, then on initiative 20, nothing prevents them from taking action (it's not their turn), but they are surprised and so they can not take action air.
A result of this literal reading of Surprise is that any creature can take action while being surprised as long as this action is not in their turn. That said, the surprise explicitly forbids reactions up to the after your turn, and the bonus actions must always be your turn.