In such dangerous situations, any good policeman would sharpen all their senses to make it out alive.

You could have also said:
"sharpen all of their senses"

However, Twitten fell into his own category.

I'm not sure what you mean with the word "category" here. Did you instead mean a word like "habit"?

He was a reflective man who greatly valued logical reasoning and expected others to do itso as well.

That’s why he was eager* to explain to Mrs Groynes how he had worked out that she was not only Crystal’s murderer but also a criminal mastermind.

She’d be ready to correct him ifhad he’d got** something wrong.

Twitten started by saying that at firstinitially there were minor details that had struck him as odd.

Your sentence is fine, however I prefer "initially" to "at first" here because it reads a little more clearly. Also the usage of "had struck him" isn't really necessary.

For example, she certainly didn’t speak as anylike a woman of her age and social class would doclass.

"social class" is a fine word to use here, but I think just writing "class" flows a bit better with the rest of your sentence.

Then, she happened to be around crime scenes just in time to “witness” the crimesm or to provide first assistanceid.

What had probably had given her away the most wahowever is when she tried to talk him out of looking into the “Aldersgate Stick-up” angle.

Only when he connected it all together, he was he able to see the whole picture.

Pretty good sentence. Maybe the English expression "connecting all the dots" may be of interesting use here, as a suggestion.

*Would it be better to use one of the following: “itching” and “craving”?

Eager and itching would both make sense to use in that sentence. I would say that "craving" may be a little strange to use and I would avoid using it in that context.

Between "eager" and "itching", I guess I would say that itching is quite a colloquial way of referring to what you may want to say with something like eager. Since your text seems to be written in the third-person, I would say that "itching" may be a little inappropriate here, and I would rather use "eager."

However both are technically fine.

**”Got” is the British spelling of “gotten”.

I've lived in England for around a decade and I would say that both are used pretty frequently. It usually just depends on what sounds better or is easier to say in a sentence.