Фандорин сделал Елизавите предложение, и она согласилась.

I think it is better to have comma here, because two parts of the sentence have different subjects.

Он положил коробку на столе и отдал Фандорину письмо.

Он закричал, и экипаж уехал как быстро можноможно быстрее .

Фандорин сразу узнал: что-то было не так.

Alternatively:
Фандорин сразу узнал, что что-то было не так.

Instead of "узнал", I would use here "понял" (this would sound stylistically neutral) or "осознал" (this is a little bit more dramatic, but not too much, you can easily see this verb used in a normal speech).

Он слышал женский голос за ним и посмотрел назадпозади и оглянулся.

My comment here is very-very circumstantial. Grammatically, your sentence is correct. There is a little issue however. In Russian, word "экипаж" is masculine, and Fandorin is, of course, too. So, when you write "за ним", "behind him", it is not clear, whether means behind Fandorin or behind the carriage (because in Russian "он" or "ним" can refer not only to human beings but to objects as well). Obviously, native speakers often do similar mistakes and there are plenty of jokes about this sort of ambiguity. I fixed the phrase to avoid it. But, again, your phrase is correct grammatically.

Он увидел свою жену в окне и в следующий момент быпроизошёл большой взрыв.

You can leave "был" there, and in a spoken language it is very common. For a written text, I would use a little bit different verb which is simply more often used with "взрыв". It also lacks some sense of continuity which carries "был". The explosion is probably a short, time restricted event)

Он осмотрелся и увидел разбитые стёкла вездеповсюду.

Both your and mine options are valid. For me personally, my one feels a bit better (that's why I suggest it after all), but other native speakers can disagree, I suppose.

Он нашёл письмо, которое он только что получил от курьера.

As an option, in many cases, when you have the same pronoun which appears in description of sequence of actions ("Он нашёл", then "он получил"), you can omit it in all cases except the very first one. However, it is not a mistake to keep it. Basically, you can just try to recall this when you encounter similar structures, for example, in books. They are very frequent. This might help to create some kind of intuition, because there is no real rule here, I think.