-How does the unexplained (crucible) interact with the unknown(catalyst). Could be important, because it´s part of the setup leading to the question.
Hackett actually tells you at some point that their engineers (I think it was them) believe that the energy created by the Crucible is able to defeat the Reapers, so they must have dug into this energy topic at least a bit, maybe tested some things out.
Their question left is how to target the Reapers and only them, and that's where they think the Catalyst comes into play.
It's not like they are completely in the dark about how it works. Maybe the details are missing, but the basic understanding is there, I'd say.
The thing is, the other Cycles clearly knew more about everything, so we just might not be as advanced. The current cycle has no other choice anyway than to try this device, if they didn't they had already lost. It's said many times that this is an act of desperation.
In the long run, yes, maybe they could've studied the Crucible in more detail, like the Protheans apparently did, but then the longer they take the more people are harvested and the more people could be indoctrinated, like it happened with the Protheans.
-How could we actually build the crucible and modify it and it´s still working?
The important part is not the Crucible, but the Catalyst or Citadel in this case. Yes, they made modifications, but in the end only the energy part and how it will be dispersed is important. All modifications they made were mostly other things that didn't have to do anything with the energy the Crucible would generate, like for example fuel pods, kinetic barriers, radiation shielding, basically everything that could ensure a longer survival of the Crucible.
-Why is destroy totally acceptable when it´s the only option?
What do you mean here? Like in Low EMS? Because the Catalyst was changed by the Crucible and has to accept the new solution? It is more hostile towards you when only Destroy is available, which shows me it has to present you the option, but it also can't prevent you from using it.
-Why did you actually let me up here?
Maybe it was not the Catalyst's doing?
This is a question I've tried to answer myself over and over again and I can't find an ideal solution, it's rather meh, but what if there was some kind of chain reaction once the Crucible docked, sending its energy to this platform too to activate it in some way.
The other possibility is that the Catalyst is forced to act on this as well because a new solution must be found due to the Crucibles' docking. Other than that, I have nothing, haha.
Either way, this is not really satisfying, but maybe a very weak explanation.
And we´ve seen no real examples of AIs going genocide, only organics.
Well I could argue the geth would've been on their way, but I won't. 
To me, this is exactly the theme the player has to look at with their own eyes. The Catalyst may present you its "facts" from all the years it existed, but you as the player have to decide if you want to believe this. It's your job to take its words for what they are and come up with your own reasoning. I saw many times that players used "but I just made peace between the quarians and the geth! I proved you wrong!", and their reasoning for the end choice is then driven by this.
Like angol said, all reasoning from players is valid. There is no right or wrong choice.
And about the trusting. What do you do in real life? If someone presented you "facts" and you had to make a choice, you would also not know if you can trust this. All you can do is follow your gut and pick an option. It doesn't really matter if you believe this person or not, because doing nothing results in nothing.