OK, here goes! If I don't quote a bit, I haven't found any errors in it. Again, it may look like a lot here, but most of these are very minor errors that are barely noticeable - and many of them are just optional, just what I think would "sound better".
I KNOW already that there is a lot of grammer errors. Please feel free to comment them, as your comments help me learn.
There
are a lot of
grammar errors. "There is" becomes "there are" if the thing is plural.
Please feel free to comment
on them, as your comments help me learn. In English, for some reason, you comment on something, you never comment something (unless your programming, but that's a whole different topic

). I'm not quite sure why, but there you go - English is a strange language.
Every time he woke up just before hitting something, he only ”knew” was there, but haven't yet seen in his dream.
That should be "
hasn't yet seen", because it goes "he has", not "he have".
The dreams began to get on his nerves.
That should either be "were beginning to" or "had begun to" - if it's "began", it sounds like they just began to get on his nerves at that point in the story (that is, specifically after he had lit his smoke) - unless that is what you meant. I have no idea what the official grammatical name of "were beginning to" would be, but I know "had begun to" is the pluperfect (or past perfect) tense, which is used to talk about something that's happened before what you're talking about, when you are currently talking in the past tense. For example, "I walked into town. Before that, I had got dressed and had breakfast". You form it with the word "had" followed by the past participle of the verb.
And they didn't progress from the first time he had those dreams.
Again, it would be a good idea to use pluperfect/past perfect here - "from the first time
he'd had those dreams". Though unlike the previous example, this one still means the same without. Out of interest, when pronouncing something like "he had had", you usually give much less stress to the first "had", so it sounds like "he hud HAD" - or, of course, you just use "he'd had".
Seemed so long ago when he said goodbye to her in the RDA complex. That was the last time he saw her. He was a soldier, back then. Fighting for reasons, he honestly didn't understand, and didn't care for. Orders were just that. And they were to be followed without question. When his sister left earth, he decided that he was done with orders. Done with the killing of people, and risking his ass in the process. Ironically, he got a job at RDA, when he decided to leave military life and start a new life as an engineer. The same company that ”took” his beloved sister away from him.
The pluperfect is only used to establish that you've gone back in time, after which you normally would change back to ordinary past tense (though staying in the pluperfect would also be fine, it seems more natural to me to switch back to the past - some people might disagree). So, if you wanted to integrate the pluperfect into this, you would only use the pluperfect in the first sentence (which you don't HAVE to do as it is obvious what you mean, but again, it's a good idea). So it would be "when he'd said goodbye to her", and the rest would be as it is now. It's good that you reinforce the fact that you've gone backwards with "back then", for example.
As long as he could remember, she had been doing research, as the scientist she was. Many thought of her as a ”nerd”. Never had time for anything else. But she had always been there to tease him, make fun of him, kick him, and talk to him when he was down. Things that siblings do. But now? Now she was too far away to be there for him... And he really needed to talk to her. He needed her to drag him up, from that hellhole others referred to as his life. Or what was left of it.
This is also good - you've already used the pluperfect in the first sentence of this paragraph, and in the sentence before the one where you go forward again to when the story is taking place, but nowhere else. I would say that is what feels most natural for me. So, no errors here.
And they still didn't know what was wrong, other than his lungs was in a bad state.
His lungs
were in a bad state - lungs are plural.
Again it was hard to breathe, and he felt that burning sensation of fire in his chest, that send him down on the floor last night.
The past form of "send" is "sent". And again, you could optionally use the pluperfect here (but again, it's obvious what you mean because of the "last night") - that would be "that'd sent him down on the floor last night".
despite she was in her late thirties.
Despite takes a noun as an object rather than a whole clause, so you would do something similar to what you do in Na'vi in these situations: "despite the fact that she was in here late thirties", or, if that seems to formal to you, just use even though: "even though she was in her late thirties", and skip the whole mess with the word despite.
They just completed the project
Pluperfect is strongly recommended here - "They had just completed the project" - otherwise it sounds like it actually happened right there in the narrative.
He's should be a ”god” in lung diseases I've been told.
This doesn't seem to make much sense to me. You can't have "is" and "should be" together - use one or the other. Though personally I think you should use "is" here (He's a "god" in lung diseases), otherwise you're implying that he should be, but he actually isn't.
Hmm. Yeah. Guess he was 'available' when the economical situation demanded the hospital shutdown.
If you were intending shutdown to be a noun here, use "demanded the hospital's shutdown" or, better (in my opinion), "demanded the shutdown of the hospital". Alternatively, if you intended to use shutdown as a verb, first of all I would separate it into two words (shut down), and you should also add a "that" after demanded - "demanded that the hospital shut down". Shutdown as a verb seems to be only used in computing, and then only occasionally. Most of the time, if it's a verb, you should use "shut down".
Although many probably had figured it out by them selves.
Themselves is one word, otherwise it doesn't make much sense.
Those were but a faint fog in Jim's memories of the PR propaganda, RDA had managed to pull off almost a decade ago.
There should probably be a "that" in there - "the PR propaganda that RDA had managed". Again, good use (and disuse) of the pluperfect in this and the following few sentences.
She knew what happened back then.
This would sound better in pluperfect, but again, it's not necessary in this case because of the "back then".
I wasn't 'leftover' from anywhere, except my education.
Leftover, like shutdown, is the noun - when it's a verb like in this case, it should be "left over".
When she met him, almost 5 years ago, he was all laugh and fun to be around.
Again, pluperfect would be better here but not required because of the "almost five years ago" - "when she had met him".
and his ability to make her mood chance to the better when she had a bad day.
You usually say "change for the better" rather than to, although you use "to" (or "into") for most things other than "for the better" and "for the worse". Again, just plain weirdness of the English language

There - that's everything I could find, and most of those are optional changes.