He was rambling through the forest
to ramble:
: to walk or go from one place to another place without a specific goal, purpose, or direction
: to go from one subject to another without any clear purpose or direction

Russian: прогуливаться, шляться, бродить

I'm not sure that 'rambling' fits here. He had a goal, after all - to find shelter, clothes, food, etc. I'm not sure, but just 'walking' suits better.

Peter could not identify some of the trees; they looked totally alien to his native California
I pointed this out earlier - this sounds weird. Grammatically you compare trees with California.
"they looked totally different from the ones he grew accustomed with back home in California"
Also, I should probably mention that birch is not 'totally alien' to North America. They appear more frequently in Canada, but certainly Peter should know it, if not by name, but by sight anyway.
Here's the birch tree areal:


with white bark stained with tiny horizontal strokes and black grainy marks.
Stroke - generally it's an action, not a crack in tree's bark.

Always use native MONOLINGUAL dictionaries - don't ever trust bilingual ones.

The weather air was rather fresh, even cold
a few young flowers
few is a noun here.

That was the last thing Peter had ever expected, to find himself all alone in the forest, without shoes, clothes, phone and other things that normally surround us and make us feel safe and protected.
English punctuation is a very tough topic, but a general rule states - if you're not sure then don't use a comma
Finding himself alone in a forest without shoes, clothes, his phone and other things that normally surround us and make us feel safe and protected was the last thing Peter had ever expected.

Plot:
He avoided thinking about his former wife who appeared not to be interested in his business or his worries or his happiness.
Did HE left her or did SHE left him? He was rich, after all and could afford such a wife. You can leave a hint or two about the history of their separation. Perhaps his inability to have children woked its part as well here.

He loved children
Loved is a bit strong word here. Try 'He was fond of children'.

The doctors had managed to recover his body
It's a simple narration, why past perfect?

He had tried to cure it but the doctors told him there was no chance.
Same.

The more successful grew his business his business grew the less happy he was
The word order is kinda important in English.

Plot:
to live a new life where he could avoid those stupid mistakes
Which mistakes? He didn't make any from what you told us. The only bad part about his life was that he never found the real purpose, but that can hardly be called a mistake.

a sudden strike of pain in the chest, a scream, a choked groan, his gasping for air and … darkness.
So many times he cried for help, but with no luck.
Too poetic. Leave it as "He cried for help many times, but with no luck."

His skin wasn’t the wrinkled skin of an old man
his muscles regained the now-forgotten strength, the old scars (no comma) which had always reminded him of the road accident (no comma) were gone
and a lock of his previously gray hair was again dark brown, as if he was young again
How can you see a lock on your hair without a mirror? If you insist then add that he found some water and looked at his reflection (in the morning twilight, yeah ) But then, if there was enough light he should have examined his face and find out that he's indeed a young person again.

He decided not to wrack his brain mind around this new state of his body because there were a few more immediate things to do, so he jumped to his feet and resumed his rambling walk.
a small hamlet village, surrounded with woods, and another road twisting and turning that twisted and turned through the whole settlement and ultimately disappearing behind in the trees on its other side
Hamlet is too specific. Village would be better.

Plot:
Strange, but he didn’t spot any cars or other vehicles and the whole hamlet looked very poor and ancient, just like it would have been looking at the first half of the 20th century.
He must have spotted cattle (cows, goats, chicken, etc). And also people - village people are early risers. He watched all day and didn't see anyone. Village lives either by cultivating the land or by its cattle.

holding something like a shirt in his hands
But pants? What about pants? And footwear?
He returned something like ten minutes later and wordlessly handed Peter a rough linen shirt and wide pants . The cloth was undyed, seemingly handmade but it would get him dressed at least.
The man also provided him with a pair of old boots, but Peter decided first to tend to his foots so he didn't put them on.

Plot:
He laid a big wooden spoon next to the plate and jerked his head at the food.
We're speaking about a Soviet village in 1941 (mid 20th century). You describe it as if it was from 18th century. I'm practically sure that there were metal spoons. Many villages had electricity by this time (not in the houses, perhaps, but on the streets at least). Also Peter was bound to see at least one car (truck) or a tractor.

Peter took his seat and started quickly gobbling the food up as he hadn’t had any meal since the morning.
The soup didn’t look, neither smell nor taste familiar
I'm pretty sure, you can't use neither...nor here. Rephrase:
The soup looked, tasted and smelled unfamiliar.

Peter was sitting there investigating the new dish kind of meal. Anyway, it was food, a fuel for the body, by this sole fact worth being eaten up, so Peter was eating ate without hesitation.
A dish is a big plate on which the food is served (посуда) not the food itself. Stick to MONOLINGUAL dictionaries.

‘You are in the Soviet Union, Belarussia
Nobody used the word Belarus till 1991.
Plot: I think the most natual answer will be the name of the village first. (Look at the map of Belarus and bind his location to real geography of the region. This way you won't need to invent geographic impossibilities further ). (деревня Букашкино, Первомайского района, Колхоз "Светлый путь" ). Then, after Peter would have naturally asked 'Where is that?' Ann should be somewhat surprised and tell him.
Also Ann would never say '70 miles', she would say 'about 100 kilometers'.

Plot:
It's June, 21st, right?
at least he was the President three months ago when I last saw your newspapers
later:
That was the date of the issue. June 12 , 1941.
It should be March, not June.

Plot:
He was woken up in what felt like the middle of the night, although it was starting to brighten outside the window
I'm being a tiresome old bore, am I?
It's june solstice. Here's the calculation for Minsk (June, 22nd, 1941): Full sunrise at 4:38. The sky was quite bright when he woke up.

They all went out of the house and all at once couldn’t help but to raise their heads up to the sky.
These planes couldn’t be anything but bombers.
These planes had no jets, but had propellers on their wings and looked very old, as if they had been built back in the WW2 era.

Nikolai took out of the old frayed cupboard a huge bottle a huge bottle out of the old frayed cupboard
(word order)

handing him a piece lump of bread
He gratefully took it and made a few bits feeling that the fire inside dies dying away.
The man and the woman Nikolai and his wife couldn’t sleep either
When Peter opened it he saw Ann standing at the door with a seal of sorrow on her face.
Не верю! (c) Станиславский. These people had not yet fully realized what happenned. They had not known yet how long this war would last and how many would die. So there should be no sorrow. They're afraid and worried. So - 'with a worried and slightly frightened look on her face'.

we saw planes and heard shooting
They could only hear distant explosions, not gunfire.

Peter felt like he wanted to give her a hug and try to soothe her pain
To comfort her, to calm her down, but certainly there was no pain.

Phew... now I'm ready for chapter 2