He lieday down as comfortably as he could, with his elbows resting on pine needles and the barrel of the machine gun laying on the trunk of a fallen tree.

lie, lay, has lain → takes no direct object
lay, laid, has laid → always takes a direct object

The enemy officer would arrive by following the horses’ tracks and he would be less than twenty meters away from where Robert had positioned himself.

The enemy officer was lLieutenant Barrendo.

Barrendo and his men had overexerted their horses andso they had had to mtake a detour when they arrived at the damaged bridge.

They had had to cross the gorge higher up the mountain and then go down again across the thick forest.

Their horses were sweaty and exhausted, to the point thatwhere they had to be forced to move on.

His machine gun was resting on his saddle, underneath his left arm.

He waited until Barrendo reached a spot in which the sun shone the brightest because there was a clearanceing in the forest.

Robert could feel his heart thumping against the pine needles on which he was laying.

He lieday down as comfortably as he could, with his elbows resting on pine needles and the barrel of the machine gun laying on the trunk of a fallen tree.

The enemy officer would arrive following the horses’ tracks, and he would be less than twenty meters away from where Robert had positioned himself.

The enemy officer was lLieutenant Barrendo.

In English, titles are also capitalized.

He had arrived from the farm and had been ordered to close in on the soldiers thatwho were taking shelter near the top of the mountain.

For people, use "who."

Barrendo and his men had overexerted their horses, and they had had to make a detour when they arrived at the damaged bridge.

They had had to cross the gorge higher up the mountain and then go down again acrossthrough the thick forest.

"Across" seems less idiomatic. My guess for why would be that "across" is used more for situations where the journey is relatively easy.

Their horses were sweaty and exhausted, to the point that they had to be forcedforce was needed to get them to move on.

This is technically correct, but I would reword the last bit as "had to be forced" sounds vague in the context of working animals.

His machine gun was resting on his saddle, underneath his left arm.

This doesn't quite make sense to me: you would assume that his hands are holding the reins, but "underneath" here implies that he is holding onto the gun.

He laied down as comfortably as he could, with his elbows resting on pine needles and the barrel of the machine gun laying on the trunk of a fallen tree.

lied is not telling the truth. When you get down in a horizontal position (past tense) it is 'laid'. If you are are already horizontal on the floor the past tense is 'lay', present participle 'lying'.

The enemy officer would arrive following the horses’ tracks, and he would be less than twenty meters away from where Robert had positioned himself.

At that distance, it would be impossible to miss.

The enemy officer was lLieutenant Barrendo.

Lieutenant Barrendo is a name with title so Lieutenant has a capital.

Their horses were sweaty and exhausted, to the point that they had to be forced to move on.

When Barrendo arrived, he had a very stern look on his face.

His machine gun was resting on his saddle, underneath his left arm.

He waited until Barrendo reached athe spot in which the sun shone the brightest because there was a clearanceing in the forest.

He has reached a particular spot so 'the'.

Barrendo and his men had overexerted their horses, and they had had to make a detour when they arrived at the damaged bridge.

Can keep the first "had", but I think it sounds better without it.

They had had to cross the gorge higher up the mountain and then go down again across the thick forest.

His machine gun was resting on his saddle, underneath his left arm.

Is the gun resting on the saddle or underneath his left arm?

He waited until Barrendo reached a spot in which the sun shoined the brightest becaus, since there was a clearance in the forest.

I think "since" works better in this context, I'm not really sure how to explain why though.