It's been ages since I last did anything like a full edit, so bear with me and keep in mind that most of what I suggest is just opinion. I'll also try to help you clean up your sentence structure a bit.
General thoughts: The whole piece definitely reads as though English is your second language. A lot of redundant words, but also a lot of missing ones. I'll try to point them out as I go.
An Army convoy rolled through sandy Nevada deserts, leaving a large cloud of sand floating in the air behind them.
This is awkward. "The convoy rolled through the Nevada desert, leaving large clouds of sand in its wake" is how I'd have written it - it communicates all the important information, it gets rid of the repeated 'sand', and it implies the action 'floating in the air' (because that's what clouds of dust do) without dragging the sentence on by explicitly mentioning it (mind you, when I'm writing posts like this I tend to drag on. Do as I say, not as I do
The leading Humvee with tinted windows turned right, off the main road, followed by four Army trucks.
The first comma is unnecessary. As far as other edits, they'd be total nitpicks (I don't quite like how you started the sentence, but it works).
Without any signs of hesitation, vehicles passed a large sign, which stated the following: “Warning – weapon testing area – unauthorized personnel will be apprehended”. The road led to a large compound surrounded by high-voltage iron fence.
"Which stated the following" is perhaps the most passive way you could have described the sign, but there's a lot more to do with this first sentence than just that. You're missing a 'the' before 'vehicles'. I'd also redivide these sentences: the first describing the sign, the second having the convoy roll past it to the compound.
Inside the fence there was a huge main building, which was a weird mixture of fortress and office complex, barracks, fighter jet hangars and a few helipads with Apache helicopters on them.
I would carefully rethink this whole description.
Outside the iron-fence gate, there was a small, green tent that held inside a squad of five unfortunate soldiers that were in guard post that day.
You keep using "there was". It's a fine combination of words, but using it more than once or twice a page is a little excessive. Try for more active descriptions, e.g. "Outside the iron-fence gate stood a small green tent, home to a squad of five soldiers on guard duty that day." Try not to mention the soldiers are "unfortunate" until you've already revealed why - it's too heavy-handed to be effective as foreshadowing.
Drinking soda, cracking puns and playing Texas Hold ‘Em, they were trying to survive the last three hours of their turn.
This is the best sentence so far from a prose standpoint (parallelism, appropriate vocabulary, Rule of Three), but you might want to rethink the soldiers' activities, because with the exception of the puns, those are some of the least Army-like things I can think of.
As the sound of the approaching convoy grew louder, the bored squad raised their heads and looked on each other.
I'd substitute "shrugged at" or some other similar expression of disinterest for "looked on". I'd assume that in a small tent packed with five people, raising their heads is tantamount to looking at each other anyway.
“Must be the supply trucks. Two hours ahead of schedule. Evans, go check their ID’s” said the officer.
Who's the officer? I understand he's prologue cannon fodder, but you should at least indicate his rank.
Youngest of the five let out a sigh of frustration when he dropped his cards and walked lazily out, leaving the comfortable shade of the tent.
This needs restructuring, as well as a 'the' before 'youngest'.
"Private Evans, the youngest of the five, heaved a frustrated sigh as he dropped his cards, leaving the comfortable shade of the tent." But honestly, I'd drop the whole 'youngest of the five' bit entirely. It's going to stop mattering pretty soon anyway.
The leading Humvee stopped right left to him
This is confusing on first glance. Replace "right left to" with "just left of" for clarity. Also, whose left? You should clarify that.
and the tinted window of the right front door opened a bit
Okay, so it was the driver's left, but you should explicitly mention that to avoid head-scratching on the part of the readers.
“Excuse me, Sir, I need to see your ID before I can let you in the compound”, Private said, raising his hand to a lazy salute.
Is his first name Private? I'm sorry to hear that - he must have been the butt of all the jokes in school. Also, "into" rather than "in".
He then gave a look to the rest of the convoy and started to wonder: if his memory served, there was supposed to be only two trucks.
"Looking to the rest of the convoy, he began to wonder: weren't there supposed to be only two trucks?"
Whirring voice of an opening window made him turn his head back to the car. Private was going to ask about the extra trucks, but the words got stuck in his uvula.
"The whir" is better than "whirring voice" in this case.
Instead of looking face to face with a strict-looking officer giving him an ID card, he was looking at nasty-looking smile of a man with crystal-decorated blue helmet – and most importantly, some sort of weapon that seemed to be an extend of his arm directed straight to the Private’s chest.
"the" before "nasty-looking". "a" before "crystal-decorated". "extension" rather than "extend". "at" rather than "to" the Private's chest, but that one is more nitpicky than the other ones.
“My name is Gemini Man, and this is how I identify myself”, he said with a soft, silent voice
A voice can be soft or silent. It can't be both, since silence is the absence of sound.
and the thought of warning the others did merely have time to raise it’s head in poor Private’s mind, as the weird man fired his weapon.
"and the thought of warning the others barely came before the weird man fired his weapon."
A bright-blue laser beam emitted from the hand cannon and pierced Pirate John Evans’ chest, killing the man.
Oh sweet, he was a pirate too? Is there some sort of massive pirate-vs.-robot ultimate showdown later on? This is something that needs further exploration!
Also "Emerged", not emitted. The cannon emits the beam. The beam emerges from the cannon.
Private’s body fell in the ground, the look of surprise and the realisation of his impending death stuck on his young face.
"The private fell to the ground"
You don't need to mention this.
Ps. The actual chapters are gonna have more lenght.
If so, then I'm not going to have time to go through them in detail like this one, but I'll still read them through and post my thoughts.