Okay, we have an opening that you're not too sure about. You seem concerned on the flow so I will try to help you with that, and ironing out those pesky wrinkles.
Eternal life. Ever since the dawn of mankind, it sought out ways to attain immortality. Yet life as we know it, wouldn’t exist without this burning desire. Whether it’s born out of greed or fear, the result is the world we live in today.
A few things here in the opening paragraph. "Eternal life", should not be it's own sentence. Instead, replace that first period with a comma. When you say "it sought out ways", since, in my mind, eternal life was still the subject, it made it seems as if eternal life was seeking itself out. I would replace the "it" with something that lets the reader know you're talking about mankind. As far as the rest of the paragraph goes, I see a couple flow issues. When you say "Yet life as we know it..." it makes me think that there was something bad mentioned about eternal life before that sentence, which there wasn't. "Yet", is a contradicting word, used when you want to contradict a point you said earlier. I would switch the last two sentences around, that way you're saying something negative first, then transitioning to a positive. This won't work if all you do is switch the two sentences, but with some revision, it should work out better.
These preparations were vital for tonight’s
operation. But for now, she walked away, making one last round through the city with calculating eyes.
I would replace the first period with a comma, and the last comma with a period.
The gym was about to close its doors though, so they probably didn’t manage to get a ticket for the show and hoped to just catch a glimpse of the trainers on the square.
I would stick with future tense right here. When you say "so they probably didn't manage to get a ticket", it doesn't make much sense because it hadn't happened yet. I would replace "didn't" with "won't" to keep the future tense consistent.
“heej, Astrae.” It sounded much closer now, too close actually.
The "h" on "heej" should be capitalized. Also, what is "heej"? I'm not sure if this is a spelling mistake, or how they greet people here.
That's all the grammar mistakes I caught. One thing I noticed is that you like to replace some commas with periods. It didn't happen often, but try to be wary about that in the future.
She threw a rope down the city walls and secured it to the railing.
Here is where I was confused a little bit on where the setting takes place. With this sentence, I think that Astrae is on the city wall, about to jump off and shimmy down to the outside world, but this sentence has me second guess that.
“Astrae!” An all too familiar voice cuts through the noise from the babbling people surrounding her,
Is she down on the busy street, or is the city wall a popular road? I'm aware that she walks away from the rope, but where does she go after that? A little more detail on where certain things take place would help.
I think I found out why you keep getting frustrated with the flow. After reading the introduction, I think (don't hold this against me if I'm wrong :P ) you're looking at this more as describing a movie, than sharing a story. What I mean is that the introduction you use here, is commonly seen as an introduction in movies. I know when I write, I keep looking back on what I just wrote and constantly think of how I can improve it. I have to rely on someone else to tell me that it's good and that it fits in. I can tell you right now, that you're flow is good. There's not any glaring holes with it, and the story progresses evenly. In movies, it's very easy to feel the flow. It's being shown to you right before your eyes, but in writing, the flow is harder to catch because your brain is trying to paint the picture in your head and it's working harder to understand what's going on. All I'm saying is don't be so hard on yourself about the flow, it's fine.
As I wrap up my review, let's recap. Don't go crazy with the periods, commas can be better in certain situations. The setting is a little cloudy and would benefit from some fleshing out. Your flow is good, don't go tearing apart your writing looking to add more in. We have an interesting story about a girl, who is not wanted in this city for some unknown reason. I look forward to hearing back from you and the next update to the story.