I think it's best not to translate the sentences from how you would have written them in Russian. You are good enough at English to "think" in English. Just "switch off" your mother tongue while you are writing, and pretend you don't know it. Trying to do a direct translation will complicate things. My biggest mistakes in English have invariably been when I tried to translate a sentence that was "perfect" in Swedish. Expressions, certain professional terms etc. Knowing them in another language actually just confuses things. The other option would be to write out the whole story in Russian and translate it like a professional translator once you think it's perfect.

Ramil knows the finer details of English grammar like commatization better than me, and on a good day his English is excellent. You can definitely trust his feedback.

The story is interesting, it's just that I don't like the second world war
I only make an exception with some Russian films, since some of them are artistically good. Otherwise I HATE this whole period. Btw, there is definitely a film involving time travel, naked guys and world war 2. Russian, quite new - mid 2000s maybe. I forgot the name, but it's 3-4 guys digging for war memorabilia in an old battlefield. One of them reckons he's a nazi, even though he's Russian. They go for a swim and when they emerge from the lake, it's WW2!

Definitely check the situation with the consulate. I am almost certain there would NOT have been a US consulate in Minsk. The USSR kept extremely good track of what foreigners got up to and helped them long before any "normal" consular emergency would occur. Almost none of the normal situations that would cause a traveller or business person to visit a consulate could actually occur in the USSR - it was a rather special situation if you compare with today.

Also I think in order to make the story realistic, the people he meets in Belarus must not be able to speak English! Language difficulties would automatically be part of the plot, if the story is going to be realistic.
Why don't you make it so that Peter can speak simple Russian instead? I think that's actually more credible than the other way around. Make up some reason.... Hardly anyone in continental Europe could speak English back in those day. If they spoke a foreign language it was usually German or French.

And I agree with Ramil that a stranger turning up in the USSR without any documentation in those days would probably be regarded with extreme suspicion unless he had an extremely convincing reason. Maybe check with your grandparents what they would have thought, back in those days, if a random English speaking person turned up on their doorstep? Would they have taken the person in and let him stay the night, or called the authorities? I think in Stalin's days, they probably would have called! Or their neighbour might have!

Seemingly should be "it appeared" or "it seemed". There is more, but I can't right now.
However, let me compliment you on your impressive English vocabulary.