It’s usually custom for theatrers not to stage a play when its respective playwright has recently passed away.

"Theatres" is correct for most English-speaking countries, I'm just offering an alternate spelling for American English

However, the news of the murder has put the seemingly doomed play in a new light, increasing the interest oin ithe play.

What you wrote is correct, but it may be a bit clearer to phrase it this way.

Rumour has it that it could be a theatrical success.

Again, this is an American English spelling alternative.

When the theatre manager gets wind of that, he greedily thinks it a perfect opportunity which, that he can exploit to the full.

Next, behind a n "in-honour-of-the-playwright" facade*, he puts word out to the cast that the show should take place.

It’s amusing to see the manager outwardly keen on the play, given that he previously despised the play, as he regardinged it as doomed and unremarkable.

He then calls out to the caretaker, who promptly comes and leads Twitten throught, up the stairs and into Miss Sibert’s flat.

Which struckikes him as odd, since they arranged to meet up.

It’s usual customary for theatres not to stage a play when its respective playwright has recently passed away.

However, the news of the murder has put the seemingly doomed play in a new light, increasing the interest oin it.

Rumour has it, that it could be a theatrical success.

Next, behind a "in-honour-of-the-playwright" facade*, he puts out word out toto the cast that the show should take place.

I'm not sure if quotation marks are necessary, but I added them for readability... Devious theatre manager!

It’s amusing to see the manager outwardso seemingly keen on the play, given that he previously despised ithe play, regarding it as doomed and unremarkable.

Your sentence was absolutely fine of course. I just think "outwardly keen" sounds slightly wrong in my mind.

He then calls out to the caretaker, who promptly comes and leads Twitten throught, up the stairs and into Miss Sibert’s flat.

Looking around the flat, Twitten drawrealises that Miss Sibert is probably long gone.

I think maybe you wanted to say something like "Twitten draws the conclusion" or "Twitten derives from the state of the rooms" - which is very cool - but unfortunately "Twitten draws" isn't right.

Which strucks him as odd since they arranged to meet up.

I think this is acceptable, but it would might be more natural to say "This struck him" rather than "Which", because I believe "which" may be more natural almost as a kind of conjunction (sorry that my grammar is terrible, I rely mostly on intuition...)

Or you might have said, "This strikes him as odd" if this must be in the present tense, unfolding in real-time as you mentioned at the beginning.

*I meant that he used this pretext to deceive the actors, appealing to their feelings towards the late playwright.

Your intention was perfectly clear, don't worry!

**I used “over” because he travelled from Brighton to London.

Very natural and common phrasing, good job!