Under no circumstances

1a. ‘Under no circumstances are children allowed in the bar.’
Is this sentence grammatically correct?
Does it mean as follows? :
1b. ‘Children are not allowed in the bar under any circumstances.’
2.
2a. Under no circumstances don’t open the door.
2b. In no circumstances don’t open the door.
Are these sentences fine?
Thanks.

To me, the answers are:

Is this sentence grammatically correct? (Yes)
Does it mean as follows? (Yes)

Are these sentences fine? (No; I tend to modify them as shown within parentheses)

Under no circumstances don’t open the door. (Under no circumstances should you open the door.)
In no circumstances don’t open the door. (In no circumstances can you open the door.)

You have changed the verb, and therefore the meaning.

The original does not mean that you cannot open the door (it is physically impossible). It means that you must not open the door.

In no circumstances must you open the door.

However, the word order is not as natural as:

You must not open the door in any circumstances.

Yes.