How to manage that magic?

First, please be more descriptive in your titles: How to manage WHAT magic? It should have been something like how to edit a 3 camera shoot, or edit a dance.
Yes, that "magic" is one of the most basic things within a non-linear editor... assuming you were wise enough to have all 3 cameras running the entire time.
Capture the video first, and import it into your project using the capture window.
Make sure your timeline has at least 3 video tracks. Drop each of them onto one of the 3 tracks. Now comes the hard part - syncing them. I usually use the audio thumbnail to get it pretty close for where audio peaks were, then zoom in to fine tune it.

Once that is done, you can use the razor to select cut points, and transitions to do the fades. I usually do a razor where I want the change, drop in a transition centered on the cut, and just extend the ends of the clips to either end of the transition.

You WILL have to "cut" out the parts of the video when you are transitioning. However, this is not destructive, and you can drag the ends of the clips to adjust them, not to mention the great undo menu that allows you to undo anything you may regret. I will usually rough cut the clips (razors on basic points where I want angle changes). Then, I cut out the clip parts I don't need, leaving a timeline with only one clip shown at any one moment. I will then go back and add transitions and tweak the cuts to the most effective moments.

I don't know what you had to use snippets for (maybe cinelerra? which for some reason doesn't have previews of its snippets). However, most middle to high end style editors use the timeline centric approach, instead of the clip centric, which is what you need for your project.

I wonder, is there a way (currently or planned) to synchronize the clips by timecode or by using clip markers?

From my experience with Premiere Pro, there you can either:
- sync by timecode, which is pretty straightforward when you have synchronized timecodes between the cameras prior to recording.
- use the clip markers method, which is easier when you don't have time code. basically you preview each clip in the clip monitor and insert a marker in them where the synchronization event occurs (i.e. a loud sound, a signal on the video, etc.).
Finally, you select the clips in the main timeline and you'd choose "Sync clips by Timecode" or "Sync clips by Marker" from a right click menu, depending on the chosen method. The clips would be now in sync.

And, just for the case that somebody implements sync by timecode, it should have an "offset" option, in frames or milliseconds. (I guess ms would be better, because frame rate neutral.)

Because I do not know a good way how to sync the clocks of different camcorders with (sub)frame accuracy.

It would be also nice, if the offset could be applied track-wise: if you reserve one track for each camcorder, all clips on that track will need the same offset...