Do questions become unclear if the asker rejects the answers given?
By definition, no; rather, they lose some false clarity. Without the rejection people have apparently mistaken the question to mean something it didn't. The question as-is may or may not be unclear.
How much weight should authorial intent of the questioner be given in these cases?
It means the world, and at the same time nothing at all.
That is, a question author is entitled to an answer regarding the issue s/he is facing (to the extent they are entitled to anything). At the same time, if a question, as asked, has an answer according to some interpretation which people feel is useful, then regardless of what the original question' author wants - that question + answer combination should exist for the benefit of readers overall.
So essentially I'm saying the question should probably be split off in some way or another.
Does the questioner wanting a different answer mean that they necessarily asked the wrong question?
Let's say they failed to make a clear enough distinction between their own question and another pertinent question for which they got an answer. (Albeit not necessarily pertinent to them).