There are quite a few protective measures in place, and the new Code of Conduct is a lot more protective than the old 'be nice' policy. First I am going to (try to) answer the questions (to the best of my ability) in order, then give some more possibilities that might have not come up.
Does moderation somehow control those who have multiple accounts? What about cases when someone downvotes all activity of another person?
Yes. We have measures in place to control and check that. There are also automatic measures in place, the thresholds and algorithms are kept secret-ish, in order to prevent people to circumvent these.
Strictly speaking, having multiple accounts is not against the rules. However, there are very few cases where users who have more than one account actually plays by the rules. As soon as a user casts a vote on anything twice, that violates the terms of service.
There are cases of serial-voting which are corrected automatically, see the help centre for more information.
If person argues like "wrong because wrong" and refuses to show his/her argumentation on regular basis, will this person be asked to stop by moderators? (CSE is about science, so "wrong without arguments" - not very useful comment).
If a user is annoying and you think {s..}he is targeting you in any way, use a custom flag on such example comments, so that a moderator can look into the matter. If we need more, we will find a way to contact you.
If someone says your post is wrong, but refuses to elucidate, then there is little that can be done. Oftentimes moving on and ignoring this is the most sane way to deal with that.
Obviously you can try and ask for clarification; use a ping command @<username> to notify the user. If a user is unwilling to respond, then that's basically the end of it.
Comments that don't help clarify a post might be noise, they can be flagged, and after review may be deleted.
Everybody is entitled to an opinion, can express that opinion, and use the vote-buttons subjectively. This is how the system works. We are a somewhat small community, so it doesn't average out as fast as somewhere else, but eventually it does.
If someone makes "hints", trying to make any user look foolish, will it be prevented?
By rules, does protection against such toxicity considered reasonable thing or "witch hunt"?
I have a bit of trouble exactly understanding what you mean here. The CoC basically protects all users alike from toxicity in general. The measure here is flagging stuff, so that it can be reviewed, and corrected.
I'd say trying to make a user look foolish is covered by this quite well.
Since you posted some examples, I'll try to run through them with example actions.
I'll downvote all your activity on CSE, since it's my right to vote and I can disagree with everything you think or say, maybe using multiple accounts. I'm not being rude or mean: I'm just silently disagree. I have a right. I also have a right to have access to 2+ accounts and vote there. Maybe one account is for my relative/friend/cat, and this person also disagrees with you.
That is pretty much covered above. As soon as a user casts votes on content twice, that is voting-fraud and against the terms of service. If that happens once, the system (or we) might not catch it, if it happens frequently, we will find it and take appropriate action.
Inadequate disputes, when I'm trying to win (not to find the truth, though). I have a right to ignore logic: we are not paid professional philosophers here. I can omit arguments, simply stating that all you do is wrong because it's wrong, that's why.
Everybody has a right to not use their brains. There are people in popular circles who show that on an everyday basis.
When such people are encountered, the best (and probably easiest) way is to walk away, end the conversation. If discussions get noisy, i.e. many words not a lot of content, they will be removed eventually. Some Q&A attract discussions, and we will be made aware of these. In some cases we move these discussions to a chatroom, to let the users continue their argument at a more suitable place; in other cases we'll thin out such discussions for the 'quality' comments.
Comment flagging is a useful tool, if you find some that served their purpose, just throw a flag on it.
Obviously, if the discussion gets unfriendly or even rude, flag that content.
I will publically "complain" about you, not in the way "user X is stupid and useless", but more like "good answer, at least, you haven't written about -thing- like some narrow-minded people do (hinting at user X)";
I don't quite understand that, but I could easily imagine that this is unfriendly. Flag such content, we'll delete it; and may take further action. (We usually know repeat offenders.)
What else?
Flagging is the tool to use, if you think there might be a problem. If you raise a flag, please check back your profile page to see how they were resolved. You can also use the following template URL:
https://chemistry.stackexchange.com/users/flag-summary/<usernumber>
If you have a general problem, need some advice, check the chat room The Periodic Table. Sometimes ordinary users are there and they can help you out with their experience.
There are usually a few moderators around that can be notified @<username> (Who are the mods?). Simply write them a message that you need to talk. You can also request a 'private' talk, i.e. a chat room where only moderators and you will be able to talk. Please keep in mind, that we are all human, and may be busy, so we could ask you to write a flag instead.
If it is something you can discuss openly, do it like now, and ask here on meta.chemistry.se.
If you are uncomfortable with any of this, (basically) your last resort is writing an email to the team. You can use the contact form.
