Let me begin at the end:
Why they do this then?
I mod because I want my community to be better. Good moderation leads to better questions and better answers.
I've got an impression, that despite their diamond-like position, they are in the reality oppressed.
Not oppressed; nobody's forcing me to moderate.
There are guidelines for being a moderator; I'm glad they're there -- even the parts I don't 100% agree with. They're pretty sensible and they're a great help to me. I want to be constructive and consistent and they're really handy for that.
Personally I am astonished at the amount of power moderators have to act on their own, without any need for consultation.
They are very careful before mentioning any non-public information
Yes, to protect the users whose private information we can see. If a moderator makes a mistake (makes private information public) it's almost impossible to undo - what's been seen is already seen. So of course we're careful with it, just as I'd want my own information dealt with carefully.
Once in a while they exercise more caution than strictly needed. That's the right side to err on.
They talk as little as possible. If they talk, they prefer citing rules or public information.
I talk a lot; I'm the second most prolific commenter on the site I moderate. I also spend time in chat and participate in a lot of questions on meta. If I am trying to explain why something someone did isn't okay, then yes, I cite the public information and the rules because that's what justifies what otherwise would look like opinion. I don't make the rules; I interpret them, explain them and try to make them work.
They don't like to say anything about their opinion. It seems as if they wouldn't have opinion from anything, only rules to follow.
I express my opinion regularly. Where it needs to be distinguished from the site norms, I try to make it clear what's only my opinion and what is me reporting what seems to be consensus.
This is really just being responsible and clear.
The common picture is somebody who has to follow so many rules so carefully,
Having a site that runs consistently and by clear rules is important. Moderators shouldn't be acting arbitrarily, but rather in ways that make sense to the people they serve (or at least to most of them).
that practically there is nothing what they could do on their free will.
Well, sure, not every rule I uphold nor every consensus I explain is the way I'd do it if I was choosing. Any person who is trying to maintain some kind of order while being consistent will find that happens from time to time. That said, that stuff still makes sense. When (rarely) it doesn't seem to, there's ways to discuss it with the community moderators ... and usually they explain something we might not have thought of. Sometimes they change how they do things, but that's usually best organized in the usual way via meta (such as posting feature requests).
As I can remember, the SE estimates the daily work need of the modship around 1 hour, which is a significant part of their lifetime.
The amount of time mods spend helping keep their sites running is significant and the rewards are usually a bit on the vague side (well unless you particularly like being called a Nazi, then it's a carnival).
My impression is that the life of a mod is essentially a continuous fight for making the chance of his de-modding as low as possible.
The possibility that I might one day no longer be a mod doesn't worry me at all. I'm not going to be a moderator forever; it's presently part of my life and probably will be for a while, but I could quite happily do without it. I don't need the diamond, I do it to help out.
I'd be more concerned that I might do the wrong thing because it might cause harm, not because it might lose me the ability to do some of those moderatey things -- I was already doing most of them before I was a mod, either alone (such as with gold tag privileges) or with other users (e.g. when voting to migrate), so my participation wouldn't change all that much -- I'd go back to answering more, and do relatively more time in the review queues again (with no flags to handle, that frees up some time). So I'd still serve the community if I wasn't a moderator.
Is it possible, that a mod has an idea, how can they make a site better, and he simply realizes it?
I sometimes have ideas for making the site better or doing things differently; I raise them on meta (like any other user can) and in chat as suits the circumstances.
It is possible, that a mod thinks somebody is an a***e
well of course - now and then that's going to happen
and he simply says it?
This is not okay, certainly not in public. Ideally nobody - moderators or not - should be talking about other users in this fashion in any public way, because it isn't in keeping with the "be nice" policy. That policy is there for a reason. Avoiding personal animosity for no good purpose means I get to spend more of my time on the fun stuff (like occasionally answering a question)
Or he has to wait until this person commits something which is enough for a mod warning?
It's possible - and encouraged - to offer guidance before things get to the stage of official warnings, mod messages, suspensions and so on. Even then, it's essential to avoid personally characterizing people. We talk about actions (you said this, or did that) and avoid trying to infer personality or motives. This makes it less personally affronting and makes it clearer how to change ("don't do those particular things" ... rather than trying to figure out how to act on "you're a problem", )
Moderators are expected (not just by StackExchange, but by their own communities, the ones that chose them) to behave like adults, not children. Generally speaking they try really hard to do that.
And, here is another problem. And this is that the argumentation between a mod and a non-mod is heavy, the non-mod can take some "risky" things. The mod can't.
Yes, the non-mod user sometimes gets away with things the moderator should not do or say.
The moderator - who has power the ordinary user does not - should not only be constructive and impartial and properly behaved, they should as far as possible be seen to be so. This is sometimes tricky, because moderators are people too, and they can get impatient, misunderstand things and so on.
Which results that the people can even a insult them and they must permit it.
We don't have to permit it but we can't (or shouldn't) respond in kind. Very rude behaviour toward a moderator can be treated just as very rude behavior toward anyone else -- you can, for example, get suspended for it. I prefer (and I have seen other moderators on my site appearing to follow the same general principle) not to be the one that acts on insulting behaviour directed at me (I try to leave it to the other mods to deal with) - aside from occasionally just deleting comments that are so rude I don't think anyone should have to see them. Any consequences I try to leave to the other mods. This lack of direct action to personal insults is because if I act on them it may be interpreted as acting in my own self-interest or out of pettiness. Again, moderators should behave like adults.
To me, if I talk with one, it is a much more motivation to remain polite that he is not allowed to attack back, as his mod powers (what he can't use in the actual case).
If I see them, I don't see judges. I see people with an oppressed personality.
I don't see us as judges, but judges - if they're any good at all - must also exercise restraint, must act impartially, must be consistent, must follow the rules they try to adjudicate, and must not respond to provocation in kind. The same is true of almost anyone in a public position of responsibility (such as elected representatives -- and if we don't get it, we should demand it).
In short, we should behave more like Spiderman and less like Batman. Yes, that may occasionally mean we don't say and do exactly what we feel like doing every instant. I wouldn't want to have moderators that did.
[If feel you'd enjoy that, you might find some parts of reddit worth a try. There are plenty of subreddits where moderators do 'express their personality' rather than operate by a consistent set of principles or the will of the community -- who feel free to be insulting when insulted and so on. I don't like the communities that result, but some people seem to; to each their own I guess*. I'd rather put my time and effort into a place that facilitates being constructive and makes it easier to put more energy into helping people]
*(Not all subreddits do this -- some do attempt to serve their community and try to enforce their rules in a consistent and clear way)