This story is written in present tenses on purpose (analyse it as if it takes place while you read it)

This is okay, but I think continuous sounds much better: "as if it's taking place while you're reading it"

Mrs. Thorpe clears up why Jack Braithwaite might have come back to his accomodation/lodging/cabin*lodging this evening: she had a visit from a red-headed opinion poll lady in the afternoon; she later happened to tell him that, and the effect on him was extraordinary; he was adamant this poll lady was a con artist, looking for her next target site, and if she was to break in, she would likely do it ion the same evening.

"cabin" would sound very strange. "accommodation" (with two Ms, not one) makes sense, but sounds a bit too general to me. We might say "his room" or "his rooms" if he rents more than one, maybe "his quarters", but I think the most natural here would just be "come back to the house".

When Mrs. Thorpe is finished with her statement, she offers Steine and Brunswick a sherry, s. She then opens the door to the living room, and lets out a scream of horror: the young playwright Jack Braithwaite lies lifeless on the floor.

I don't think this "the" is grammatically necessary but it sounds a lot more natural to me. And again continuous is more natural: "is lying lifeless"

Broken furniture and blood spilt across the room indicate a life-and-death struggle must have occurred here.

By the time the police trace Alec Forrester to a pub, he is completely drunk; witnesses say he has been drinking there since mid-afternoon, long before the murder of A.S Crystal occurred.

However, Inspector Steine dotakes not take account of ithis, so Forrester remains under arrest.

Sounds a lot better this way around. But very appropriate expression!

As for the Braithwaite’s murder, Inspector Steine has to acknowledge Forrester could not have physically overpowered Jack Braithwaite in a hand-to-hand combat, given both their heigthts.

Or "the Braithwaite murder". "In a combat" is acceptable but just "in combat" is much much more common.

The Inspector comes to the conclusion that the intruder must have expected the house to be empty, so he or she, being bewildered by Braithwaite’s presence, killed him out of panic.

Whereas, Sergeant Brunswick believes the red-headed opinion poll lady and the intruder, are one and the same.

At this point, Steine goes along with Brunswick’s theory, prompting that it would have be a very strong woman; they both know where to go.

I'm not sure I completely understand what you want "prompting" to express here.