Sergeant Brunswick acquaintsshares with Inspector Steine with the latest developments onregarding Joanna Carver.

Brunswick had someone around* her lodgings; she was not there and clearly long gone.

"Wasn't" is much more common in speech/writing than "was not".

They agree that their new prime suspect must bs are a couple of a man and a woman.

Before Brunswick leaves the office, Steine instructs him to tell the irregular man downstairs to leave the police station.

I don't really understand what you meant by "irregular" here.

‘"But sir, that’s Harry Jupiter of the Daily Clarion, the “'Policeman’s friend”'".

To make things worst for him, Steine is too deeply concerned about other matters to take in what Forrester is saying.

It dawns on Steine that Twitten’s suggestions on the script subjectopic of the script for the next radio talk could be rightgood.

"Good" might not be the correct word here, but "right" felt out of place to me.

Brunswick makes Forrester aware that the police have witnesses to his making threats againsthat saw him threatening A.S. Crystal.

He put these threats down to Crystal’s upcoming and crushing review against Forrester himself.

**Which of those two is better (if they’re right)?

I was kind of confused by what you were trying to get across there, but the "stand by his order" part is right.

Sergeant Brunswick acquaints Inspector Steine with the latest developments on Joanna Carver.

You can also use "briefs" in place of "acquaints". Acquaint is more to familiarise a person from a relationship point of view, but briefing is providing facts and informing someone.

Brunswick hadsent someone around to* her lodgings; she was not there and clearly long gone.

"Not there" and "long gone" do sound repetitive together, I'm thinking it's more stylistic but you could also try, "she was not there, and hadn't been for some time."

Her alibi, unfortunately, does check out.

I see this as speech but not narrative text. I would rephrase to something like, "With her alibi checked out,"

This rules her out as the con artist posing as the red-headed poll lady.

Combine with previous sentence. Also, I'm not sure who the poll lady is, without story history this doesn't make sense to me. Maybe "the red-head" is enough, or "the polling woman with the red hair".

They agree that their new prime suspect must be a couple of, a man and a woman.

"Couple" is optional, but useful description if the two are in a relationship together.

Before Brunswick leaves the office, Steine instructs him to tell the irregular man downstairs to leave the police station.

Irregular is better used for objects, not people. "Irregular behaviour" is ok, though.
A better word might be odd, strange.

I expect he writes where he wants at Scotland Yard.’ Still, Steine stands by his order**.

This works well, you made a good choice.

To make things worste for him, Steine is too deeply concerned about other matters to take in what Forrester is saying.

It dawns on Steine that Twitten’s suggestions on the script subject suggestions for the next radio talk could be right.

Forrester twists in paingrimaces at the recollection.

Or "winces".