Seemingly, he was understood and the man’s answer sounded pretty much like ‘Nikolai’.
Avoid using passive voice. Switch to English
Apparently, the man understood him. His answer sounded pretty much like 'Nikolai'.

Meanwhile the man laid an old plate all covered in scratches
old scratched plate.

Hmm, I don't want to sound overly critical, but a bit of more detailed description of what's been in the house might be in order, don't you think?

Peter took his seat and started quickly gobbling the food up as he hadn’t had any meal since morning.
Details, man! They distinguish a good fiction from trash. He was at the evening party in his house, remember? Or it was in the morning? Did he wander in the forest for a whole day? He must be very tired and sleepy by now, young body or not.

The soup didn’t look, neither smell or taste familiar, which was weird, because Peter in his old life had tried almost every single dish that had ever existed in the world
The soup looked, smelled and tasted unfamiliar. And you CAN'T eat dish! A dish is not food, it's a big plate

where the arrangers hosts were constantly trying to surprise their guests
so Peter started continued to eat without hesitation.
He already started since he knew how it tasted.

Just when Peter finished the meal, the door opened and the old woman came in, followed by another woman.
Perfect timing, heh? Did they wait by the door

they both froze as their eyes met
Yawn... how predictable.

she looked gorgeous
I don't think 'gorgeous' is a right word for a simple rural beauty. It's more fit for an urban hi-class prostitute.

- Where am I? – he held his breath, deep inside ready for an answer that he’s in heaven…or in hell.
- You are in Belarus, about 70 miles due North from Minsk – she said with a slight accent.
The correct answer should have been 'Soviet Union'. And even if otherwise, it was Belorussia, not Belarus.
1) There was no US embassies in Belarus. A consulate maybe. The embassy is (and was) in Moscow.
2) The first thing the old man should do seeing a naked unfamiliar foreigner is to bring an NKVD officer, not a pretty girl who just happen to speak English in that village. Not many people new foreign languages, certainly very few in Belarussian villages and the most popular language was German, not English.
3) Even if these people were all crazy natural curiosity would have made them to ask questions first, not providing answers to a potential spy.

- It’s late now, you better get some sleep and tomorrow we will try to help you get to the USA embassy in Minsk, I think Mr.Roosevelt will help you out. – Ann smiled.
Why did she decide that he was American? Back in that time an English speaking person would sooner be associated with UK, not USA.

There, almost unseen in the darkness of the sky which was only beginning to lighten had only started lightening in the rays of the raising Sun, were flowing they saw numerous silhouettes of airplanes, sometimes invisible, sometimes showing in the gaps between disappearing and appearing again in the clouds.
He half-filled the glasses with the liquid from the bottle and said something that could be interpreted like ‘here, drink, you may need it’, given his own and his wife’s appearance.
I don't quite understand the bold part. Do you mean 'judging by his own and his wife's appearances'?

When Peter finally fought subdued the liquid and opened his running eyes he saw the old man handing him a piece of bread.
The man and the woman couldn’t sleep either, so the whole morning they all were just trying to keep their hands busy doing something. However everything they were doing didn’t seem either necessary or important, apparently they were just trying to distract themselves from what was actually happening.
What did Peter do? You said 'they all were'.

It hasn’t been calm at the border for a month or so and everyone was expecting something to happen
A general knowledge says that it's BEEN quiet and very few actually expected anything.

Peter felt like he wanted to give her a hug and try to calm her down but he refrained restrained himself (no comma) because it might look awkward unappropriate.
He briefly ran through the headlines and suddenly something caught his eye. That It was the date of the issue. June 12 , 1941.
(issue's too official here)

- What day is it today? – he asked Ann hoping that he wasn’t dreaming.
Was? or Wasn't?
... hoping it was a dream.

P.S. A realistic finale for a foreigner with no documents at war-time is - arrest, interrogation, and execution - everything within a week. Still, I'm very curious what will happen next.