You have misunderstood how the trem is supposed to work sounds like.
Have you put the strings on yet?
Attach one spring to hold it on then install the strings but not tightened or tuned to pitch.
Install the other springs then tune to pitch.
Trems are a balance between string tension and spring tension.

Definitely do not cut deeper grooves in the trem posts. Like Richard said, the grooves in the posts are only designed to hold the trem in place when it's sitting pretty close to perpendicular to the posts. First, get the posts installed somewhere pretty close to their original height, and it'll be less likely to tilt so much out of alignment and jump out, then you can follow Richard's instructions from there.

One other thing that may help would be to put something soft underneath the fine tuners of the trem to keep it perpendicular and off the surface of the guitar while you put the springs & strings on. For a top-mount trem, you want something around a half-inch thick or so. You can use one of those rectangular pencil erasers, a folded washcloth, a pencil with a paper towel rolled around it, etc. That will take the place of the string tension balancing against the springs until you get the strings installed, and will also keep you from scratching up the guitar

Great answers, guys. I'm completely new to Charvel/Jackson and their ways, so nothing is really obvious for me, yet.

Anyway. I put on three strings and one spring, so far so good. But I have found that my string height all over the neck is zero, and the string won't make any sound. I can't understand it, because I haven't really changed anything that would effect the action.

You removed all the strings, so the neck will want to bow the other way. Get all the strings back on, tuned to pitch, then the neck should return to the proper relief/curve.

It has nothing to do with Jackson. It has to do with Floyd Rose (and any style of floating tremolo).

Well, from your original post in the thread, it sounded like you had either removed the trem posts and re-installed them, or at least had been lowering them when you were having the issue with the trem coming off. Lowering the posts is going to lower the string height. You need to get all of your springs & strings installed, and then get your posts set to the optimum height. This is tricky stuff for somebody new to floating trems.

The only Model Series guitar I have handy is my Model 6. It's neckthrough rather than bolt-on, and the neck angle might not be exactly the same, but I'm going to take a height measurement for you, and that may help you get your posts set closer to a good starting point. OK, the measurement from the surface to the top of the posts is 13mm, or right about 1/2". Try that as a starting point & see if it helps.

Plenty of great tips on you tube on how they work and how to set them up.
It can be daunting the first time you do it but I commend you for taking the task on and learning how they work.Kudos to you.
Never assume anything when it comes to guitars because they will surprise you with all the quirky things they do when you work on them.
Any guitar with an OFR trem is the same as far as setting them up.

I'm truly impressed by this reception and your support. Thank you all! It did make the procedure much easier. A lot of the aspects do seem kind of obvious in hindsight, like how the posts do affect the string height, how the tremolo system works, and so on. But I quess it's a part of the learning process. Heh.

I set the trem-posts at 14-15mm, and it worked wonders. The guitar isn't fully set-up yet, I suspect I might have to adjust the truss-rod a bit, but that should not be any problem. I experience some string rattling at the open A and D, I will take a closer look and examine what might be causing it. Otherwise, it's perfect.

Once again, thanks a lot. Here's a picture of the (almost) finished guitar.

I'm truly impressed by this reception and your support. Thank you all! It did make the procedure much easier. A lot of the aspects do seem kind of obvious in hindsight, like how the posts do affect the string height, how the tremolo system works, and so on. But I quess it's a part of the learning process. Heh.

I set the trem-posts at 14-15mm, and it worked wonders. The guitar isn't fully set-up yet, I suspect I might have to adjust the truss-rod a bit, but that should not be any problem. I experience some string rattling at the open A and D, I will take a closer look and examine what might be causing it. Otherwise, it's perfect.

Once again, thanks a lot. Here's a picture of the (almost) finished guitar.

My advice would be to make sure you do some reading and/or watch some youtube videos before messing with the truss rod. Get the basic concepts of neck relief, how to measure it, and how much you're shooting for before you even touch it, and remember that it can take as little as 1/8 or 1/4 turn to have the effect you need. Also, string buzz can have several other causes, especially on an older guitar with some wear on it.