Gnoupi and voyager cover the general case, so I'll tackle your "extra ideas". As for downvote explanations being shown to the question author only, I don't see it as being any different than the general case, a downvoter doesn't need to appeal their opinion to the author any more than to whoever is being downvoted. So I still redirect to Gnoupi.
Downvotes are an expression of the opinion of whoever cast them. An opinion does not have to be "satisfactory" to anyone in order to exist, so there's no reason the author of a question should be able to undo them. That's why, among other things, not even moderators can change votes. It's simply against the spirit of the system. This is also why the "reliability" of such downvotes is not of any concern to the author of the question. If the author wants to take downvotes into consideration, then the author should evaluate the answer personally. The author should be able to judge whether an answer is helpful to the author by merit of the answer itself, whether or not the downvotes are "reliably given".
As far as rewarding reputation for downvote explanations, there is no reason why a downvote explanation is worth more than, say, an upvote explanation, or any kind of fixer comment that isn't tied with a downvote. For example, there are often times that I will make comments, both here and on SO, which may sound like a downvote explanation (and the post is indeed downvoted), but I didn't cast the downvote. All I wanted to do was suggest some explanation why a downvote may have been, or point out a flaw in the post. And these may lead to the post getting fixed. The only way I'd receive reputation for this act, under your suggestion, would involve me downvoting, but I didn't find it necessary to cast a downvote. Is my contribution worth less because I didn't cast a downvote?
Ultimately, regardless of whether you are the author of the question or the author of the answer that got downvoted, there are two courses of action that you can take when you get a downvote without a reason, or without a satisfactory one. You can check to see if the answer is indeed flawed, and fix it appropriately. And if you don't see any flaws, then disregard that downvote, because a good answer will get upvotes from the people who responsibly analyze these things. But that person who downvoted you is entitled to their opinion exactly the same as anyone else is entitled to upvote you without reason.
EDIT
Hooboy, you've given me a lot to have to respond to. Rather than worry about fitting it in comments, I'll just edit my post body. If you want to skip past all of the lengthy stuff, do so but please go to the last two paragraphs. Those are the most important ones here that summarizes the faults in your suggestion.
Addressing comments 4-6, that this is an "extra suggestion" is the reason I wrote my answer, I wanted to explicitly address your extra suggestions. You are casting the OP in some radiant light that implies they know absolutely what is good and what is not, which is entirely incorrect. The point of votes is to show the community's interpretation of an answer. Just because someone asked the question, it does not give them any right to decide whether someone's opposition to an answer is appropriate or not, no matter the explanation for that. And there is definitely no "moral binding" here. The question asker came to get an answer to a question, not to personally evaluate the opinions of others. The explanations of others is helpful in identifying how useful a particular answer may be, but it isn't the OP's responsibility to point out what is right and wrong in votes. Why should the OP have to judge this when the whole point of asking the question is to get a right answer?
You specifically suggested that the OP be able to reward a good explanation for a downvote with reputation. Your exact quote is "maybe the OP can give a vote to the reason, and we could think about mechanisms where votes for the reasons influence someway the reputation". Since these OP-only reasons only show up for downvotes, this is very much going to reward people for downvoting and explaining rather than those who just explain flaws, so I'm not wrong there.
I agree that the OP can always benefit from explanations of downvotes. In fact, anyone who has the same problem can benefit from it. But you're casting such things in such a positive light without paying heed to the problems that arise. If, for example, only the OP can see a particular comment explanation, and someone gives a reliable sounding explanation because that user, as they understand the answer, believes there is a fundamental flaw with the answer, the OP can consequently choose to follow that advice and avoid that answer. However, that user may be completely wrong about the answer, and if the OP doesn't know this, then there is no way for anyone to explain that. So an otherwise perfectly serviceable answer can be overlooked because of "reliability". Ultimately, the reason that a downvote is an "opinion" and not a "fact" (which the fact that answers are facts is completely irrelevant) is because it can only be given in the experience that the caster has. There are cases where someone can be 100% certain that their downvote is appropriate based on their knowledge, even when the answer may in fact be useful. And the OP, who let's not forget is asking the question in the first place, may not know this. This is why the most important thing a user should do is evaluate answers. Explanations for downvotes help, but they are unnecessary, a good evaluation of an answer can identify the flaws if they exist. And if they don't exist, then you don't need to understand why one user thought it wasn't useful to know that you yourself found it useful.
You also make a faulty assumption in the end of your comment stream that upvotes are always about correctness and rightness. A lot of people also give upvotes for reasons like "It's a good thing you suggested this particular method for this part". Not all questions and answers are clear-cut one-part solutions. Or maybe "Oh, hey, I was investigating this other problem and it turns out that this component you used was helpful to me". These answers were useful to people, but not necessarily to the author. There are also upvotes for friendliness, pity, and other less reputable reasons, just the same as there are bad reasons that people downvote. Upvotes are no more sacred than downvotes are.
The ultimate thing to draw from this is, requiring comments does not actually strengthen the system. The reasons that it, in general, is not useful is explained by all the other answers here. Making it visible only to the OP also doesn't necessarily strengthen it - the OP, just like the people who cast votes in the first place, is no less prone to incorrectly understanding things. So there is no ensured benefit to this. Anyone who would give good explanations for downvotes doesn't need to be forced in the first place, and certainly doesn't need to reserve it for just the OP.
It is fundamentally better that such reasons be out in the open to all users, so that anyone can evaluate the reason and, in the fact of downvotes made on faulty assumptions, correct the comment with their own knowledge contribution. This kind of thing will actually benefit the OP far more than just letting them see one user's opinion. Which is why the score of a post is the votes cast by the community.
